I think they'd meet Thor first. Since Thor is the one who has the closest connection to the Avengers among the Asgardians. He could be waiting for them at the end of the Rainbow Bridge or he might even accompany them on their journey to Asgard. Then he'd introduce them to the other Asgardians like his father Odin and his brother Loki (if Loki is in a good mood that day).
Probably Heimdall. He guards the Bifrost, the Rainbow Bridge that leads to Asgard. He has the ability to see all, so he'd be the first to spot the Avengers coming. And he's not one to let just anyone pass without a bit of interrogation.
In fanfiction, if the Avengers go to Asgard, there could be a big clash at first. The Asgardians are proud warriors and the Avengers have their own ways. But then they might team up to face a common threat. For example, a powerful dark magic user could be targeting both Earth and Asgard. The Avengers with their diverse skills and the Asgardians with their magic and strength could form an unbeatable alliance.
One reason could be that they see Loki as a victim of circumstances. Maybe they've come to understand that his actions were often influenced by forces beyond his control, like Thanos' manipulations. So they decide to defend him in Asgard.
You can start by looking on fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. These platforms have a vast collection of Avengers fanfiction that might involve Asgard. Just use the search function on the site and type in relevant keywords like 'Asgard' and 'Avengers' to narrow down your results.
A common plotline is the Asgardian princess sister coming to Earth to seek help from the Avengers. Maybe Asgard is under some sort of threat, like a new powerful enemy has emerged, and she believes the Avengers can assist. Another plotline could be her being exiled from Asgard and finding a new home and purpose with the Avengers.
He could meet the Frost Giants. Since he has some connection to them already, they might welcome him or at least be intrigued by his presence. He could learn more about his heritage from them and maybe even form an alliance.
In this fanfiction, Tony might also meet Loki. Loki could be either a friend or a foe to Tony. He might try to trick Tony for his own amusement or they could end up working together against a common threat to Asgard.
One popular theme is the interaction between the Asgardian characters and the Avengers. For example, Thor introducing his Asgardian culture to the Avengers team members. This can lead to interesting scenarios of cultural clash and cooperation.
Maybe Loki is wrongly accused of a crime in Asgard. The Avengers, having had their own experiences with false accusations, step in to defend him. They believe in justice and know that Loki might not be guilty.
Well, aside from Thor and Odin, he could meet Sif. Sif is a strong Asgardian warrior, and she might be interested in Harry's unique brand of magic. And of course, there's a high chance of meeting Loki. Loki is always up to something, and his devious nature could either make him an adversary or an unexpected ally to Harry.
One popular type of fanfiction in this category might involve the sister of Thor or Loki, who is a princess of Asgard, interacting with the Avengers. For example, she could be a powerful magic - user and help the Avengers in their battles against new threats. There are many fanfiction platforms like Archive of Our Own where you can search for such stories using relevant tags like 'Asgard princess' and 'Avengers' to find some great reads.