Steve Rogers allowed himself half a day to analyze his entire situation. The moment he woke up with strangers surrounding him - especially strangers wearing outdated clothing - he had a good reason to be both confused and wary. He knew at once that there was something wrong. He didn't exactly know what it was until the Maester told him that he was in a place called Westeros, specifically in the North, and the castle they were currently in was called Winterfell.
Of course, he didn't believe it, not even when he looked out the window of the Maester's tower just to survey more of his surroundings, where he immediately saw the people wearing Medieval clothing and doing certain tasks. At first, Steve was more than puzzled at the odd sight, but as he carefully scrutinize his environment, he began to consider the possibility that this wasn't an illusion of Loki's making. If it were, it would have required a lot of power from Loki to completely involve a lot of people and build an entire castle for the sole purpose of what? Deceive him - Captain America - into thinking that he had somewhat gone back in time?
Other than using the sceptre in something as elaborate and time-consuming as creating a Medieval setting with people in it, Steve couldn't see Loki going through these great lengths just to deceive him. He understood that Loki was a God of Mischief, and even perhaps Deception, but Loki was also methodical and practical when it came to using his resources. The God certainly would not squander his time and power in something like this. However, it also didn't mean for Steve to dismiss the idea so easily.
Not even a day after he woke up, Steve already knew that he was not welcome in Winterfell, because many believe that he was a foreign stranger who was found by Lord Stark's daughter in the woods. Even Lord Rickard Stark openly disapproved of his presence in his home that Steve suspected that the Lord would soon forced him to leave even when he hadn't recovered yet, which was something Steve did not look forward to. Not when he planned to learn more about his arrival in the place called the Wolfswood and discover more about the entire situation he found himself in.
Hence, after two days of bedrest as advised by the Maester, Steve immediately set out to investigate his situation further. Although it took a lot to convince the Maester to allow him to go out and return back to the place where the girl, Lyanna Stark, found him. With Brandon Stark and other guards as his guide, Steve went back to the Wolfswood. The eldest Stark was the one who pointed out to him where his shield was found and where the girl had found him days ago.
It was then that Steve realized that he was not stuck in some kind of elaborate illusion that Loki had concocted. Not when he constantly tested the people around him. He observed their reaction to his probing questions and noted their interaction with keen interest. Furthermore, he surveyed his environment, inspected his surroundings closely in fact that he often used his senses of touch, sight, hearing, smell and taste just to determine if everything around him was real - although he did so discreetly - and slowly discovered that everything was real enough for him.
As he searched the Wolfswood, Steve had hoped to find some clues of how he arrived there, but he understood that it would probably take him days to scour the entire forest for some kind of portal or any magical residue that might be responsible for his arrival, and he was also hoping that the portal or the magical force, which had sent him there, could send him back home as well.
Since Steve intended to continue with his investigation in the Wolfswood, he had to find a way to convince the Lord of Winterfell to let him stay in the Castle. Because other than his intention to search the Wolfswood, Steve also realize that he needed to learn more about this world and find out ways to get back to his old world, if his investigation in the Wolfwood came up with nothing. Both of which he could not do without learning from the Maester or having access to the Winterfell library at the moment.
Thus, after learning that House Stark uphold honor and duty above all else, Steve decided that he won't mind working for the family, despite his thoughts regarding their medieval views on things. So he talked to the Lord of Winterfell and presented his case. Fortunately, Lord Rickard Stark was a reasonable - but suspicious man - and had hired Steve as a temporary Castle guard. And since the Lord already understood that Steve only planned to stay for a couple of days at best, the Lord did so by extracting a modified oath of fealty from him for his brief service. Apart from that, Steve had offered his temporary service for free as a recompense for the aid that House Stark had - and would - provide him, which Lord Rickard found more than an acceptable arrangement. Obviously, the Lord did not mind hiring an extra hand without paying him in return. Although Steve suspected that the Lord of Winterfell had also hired him in able to keep an eye on his whereabouts in the Castle, which the other guards had not failed in doing.
Still, for the next few days - often early in the morning - Steve continued to investigate the Wolfswood with a desperate hope to find the possible portal that will take him back home. There was no other reason for him to remain in that place, not when he knew there was an alien invasion that he needed to stop. However, as the days turned into two weeks and his search in the Wolfswood seemed to be in vain, Steve began to direct his attention to other ways of returning home. He was becoming a little bit desperate to find his way back home that he would try anything, anything at all to go back to New York and stop Loki's invasion once and for all, wherein he sorely hoped that he won't be too late once he did find a way back.
With the help of the Maester, Steve learned more about this world and the presence of magic. Although the Maester was very skeptical and downright disapproving about Steve's interest on the matter, yet Steve persisted in learning more about the magic of this world. However, the Winterfell library was limited to certain subjects - and magic was something of an obscure topic - that Steve discovered that even most potential Maester's didn't even delve unto it, except for those with a passion for the mysterious arts. The only way Steve could find some answers to his questions and his chance to return home, was to travel to places where mages or magicians practice their arts freely, or Steve could go to the Citadel and pay the fee - which he heard was quite an expensive sum that only the wealthy could afford - to gain access to the general library, where wealthy merchants and nobility are usually allowed to enter.
However, Steve Rogers was neither wealthy, nor was he noble. To most, he was just a foreign stranger with almost nothing to his name, except his suit and his shield, which he didn't plan to sell in exchange for money. So gaining access to even the general library in the Citadel would be terribly difficult for him. Even though the Maester was willing to send a recommendation letter to someone in the Citadel so that Steve had the chance to enter, but that won't be enough as the Maester had told him. Steve still needed a lot of money to pay the entrance fee and another trustworthy person - another Maester or a Lord - to send another recommendation letter to back him up as someone important enough to gain access to books that the Citadel deemed as more precious than gold or Valyrian steel.
For that reason, when Steve saved Lyanna Stark from her abductors, he immediately took this chance to ask the Lord of Winterfell a favor after the Lord had offered him a job. Of course, Lord Rickard had warmed up to him at this point and had been open to negotiation. Hence, the first thing that Steve did was explained himself and his reason for wanting to work in only six months or even less as Lyanna Stark's sworn shield. Something that seemed to shock the Lord of Winterfell to no end. Steve, of course, carefully explained further to the Lord of Winterfell that he needed to return back to his homeland, but he would need money, resources and a lot of information to get back there. That's when he asked the Lord about the recommendation letter to gain access to the Citadel.
However, that was also when Lord Rickard pointed out that the Citadel won't easily accept some commoner in their special library for nobles and the wealthy. Lord Rickard explained that Steve might need a title or some kind of wealth to his name. Fortunately, it was the Lord who remedied the matter and had told him that he would knight Steve, for the main reason that Steve saved his daughter and because the Lord suspected that Steve - a foreign stranger - was some kind of warrior of renown in his homeland.
Nonetheless, for Steve to be knighted, the Lord set out a series of tasks that he needed to accomplish within a week and a half before the Tourney at Harrenhal. Most did not know about the test that Lord Stark had set up for Steve two weeks before they were about to leave for the Tourney, not even his sons or his daughter, who had been with Steve when he had started practicing with the sword and running back and forth through the Quintain. Although the entire household was aware that there was something going on when Steve began shadowing a knight in Winterfell. However, Lord Stark did not explain this to anyone.
When the set of task started, Steve, of course, did it without fail. He was only asked to squire for a couple of days with a Winterfell knight just to learn the basics of knightly duties. Thereafter, the Lord of Winterfell tested his battle prowess - with different weaponry - and against multitude of Castle guards and some Winterfell knights just to prove that Steve was a man worthy of a knightly title. No one would question the Lord's decision on the matter at all after Steve Rogers defeated most of his challengers. If they dared question Steve's new title, they would have to ask the Lord of Winterfell and those who fought Steve.
Thus, Steve gained a knightly title before the Tourney, which he earned with respect and Lord Rickards blessing. It's only a few days before they leave for Harrenhal, when it was announced to the Northern party about Steve's new title and it seemed most reacted favorably to the news. The people of House Stark already suspected that Steve was someone noteworthy and honorable in his homeland that no one dared question Lord Stark's decision for giving him the title.
Even though most assumed that Steve was going to work as a sworn shield for an indefinite time. However, no one knew about Steve's and Lord Rickard's condition on the matter. The two men had agreed that Steve would work as Lyanna's sworn shield for six months or lesser than that while Lord Stark had been adamant to keep this condition between them. Since what Steve was asking was highly unusual and some people might even question Lord Stark's decision in hiring someone who only planned to work for less than six months, the Lord of Winterfell wanted to keep their negotiation a secret, until the time Steve Rogers was ready to leave House Stark's service. So mentioning Steve's brief contract to anyone would prove unfavorable to both of them, most certainly when Steve just recently received his knightly status.
Despite Steve's intent to serve in a short period of time, the Lord of Winterfell was a generous and good man. Other than having him knighted by a Winterfell knight and giving him his blessing, Lord Rickard had easily given him the recommendation letter for the Citadel visit and had constantly advice Steve on matters that might help with his situation. The older man was the one who advised him to join the Tourney for the chance of earning more money that Steve could use for his future travels and the Citadel entrance fee. This would also mean that Steve won't need to stay for six months of service for the wage that he intended to save. If Steve won the Melee fight and the jousting, Lord Stark had pointed out that Steve might only need to work for two to three months after acquiring the monetary reward.
Steve, of course, had already considered joining the Tourney before Lord Stark's advice. Since he knew that if intended to travel in search for his way back home, he won't be able to avoid encountering certain malicious individuals who would do him harm. For this reason, he wanted to join the Melee with the intent to first gauge and test the people's fighting style in this time period to get a better idea of what he was about to expect in a real fight with deadly opponents in his future travel, and perhaps improve and adjust his martial skills ahead of time to counter his adversaries moves with either his shield or a sword. Other than that, he wanted to see what a Melee fight was like and he was curious to see how he would face multiple seasoned fighters all at once who were using medieval weapons. This was certainly something he hadn't experienced before during his time.
As for the jousting part of the Tourney, Steve realized that he wouldn't mind learning more about handling a lance in a joust, which was another skill he could add to his expansive repertoire. However, he was mostly interested in the jousting because of the reward at the end of it. He still needed the money for the entrance fee to the Citadel's general library. Hence, he didn't hesitate when he signed up in the list for the Tourney for these reasons alone.
The other thing that Lord Stark advised him was to not only carry his shield, but a sword as well just to keep up with appearance and blend in well with other notable warriors. Steve understood that most of House Stark's household already accepted his strange abilities and his mannerism, where majority of them had dismissed his oddity because of his foreign origin. Nonetheless, Steve knew that many of the outsiders would question his methods and would probably challenge him if he only brought a shield as his weapon. And despite being good in hand-to-hand combat and wielding his shield, he also realized that he might require a lethal approach in facing far more dangerous enemies in this world once he started travelling. So Steve Rogers had started carrying a sword at his hip to avoid unwanted questions. Although he would probably used a sword or other lethal weapons as a last resort if it came to that.
As Steve Rogers thought back all those weeks ago when he first regained consciousness, he was grateful that he had not acted foolishly when he was making his decision regarding his situation. If someone else would have ended up in his shoes, he was certain that they would have given up their search on the Wolfswood and began their search elsewhere at once, perhaps in other places like the Citadel or other magical places without even planning ahead or considering things first hand, like the money, important documentation or supplies, which were all vital things for travel and gaining access to an exclusive place, such as the Citadel.
Steve could just imagine some fool travelling to certain places without acquiring these and simply start taking odd jobs here and there but only to be denied by most people because of suspicion. After all, this world was teeming with cutthroats, hired mercenaries, thieves and bandits on the road that the common people would certainly be ambiguous about hiring a passing traveler. This was not the '60's in America where anyone could travel across the country and earn money doing odd jobs like the good old days. This was a world where the Mountain Clans, or the Cannibals and the Wildlings Beyond-the-Wall exist. Only a fool would assume that this world was the same one as Steve's own world, and Steve was no fool.
After the Tourney, Steve decided that he would continue with his investigation on the Wolfswood while he saved more money and acquire supplies for his future travels. Once he accomplished that, he had every intention to leave Lord Stark's service and search for a way back home. He could only hope that he won't be too late when he got back to his own world.