I'll give a complete rundown of the main Young Avengers stories first. The Young Avengers have a sporadic publication history, which is kind of a pain to track, so let's lay it all out in order. I don't have Marvel Unlimited, so I can't confirm if all these are available there. Given the age of most of these, I'd expect them to be.
- Young Avengers vol 1 (2005-2006) (This is what you've already read.) - The team's origin story.
- Civil War: Young Avengers & Runaways - A short cross over with the two teen teams, set during the Civil War event.
- Young Avengers Presents - I haven't read all of these yet personally, but my understanding is each issue focuses on a different member of the team. Each issue is a short story about the individual in the greater Marvel Universe. The one I read is Hawkeye's, which has her interacting with Ronin (Clint Barton).
- Secret Invasion: Runaways/Young Avengers - Another cross over with these two teams, this time occurring during Secret Invasion.
- Dark Reign: Young Avengers - During the Dark Reign event, a new Young Avengers team forms, and comes in conflict with the original Young Avengers. I haven't read this one personally.
- Avengers: The Children's Crusade - The Scarlet Witch returns, and the Young Avengers are one of many factions seeking her out.
- Young Avengers vol 2 (2013-2014) - This is the longest Young Avengers story since Vol 1. 15 issues that tell a self-contained story of the team dealing with a powerful evil that they accidentally unleash. It's part of Marvel NOW, and is what's currently easy to find on store shelves.
With the exceptions of #3 and 5, which I haven't read, I recommend reading all of these, as they all offer excellent stories involving the Young Avengers team. The must-reads are #1 and #7 though, so if you want the basics, those are the key ones, and they're the more substantial stories. The rest of these are shorter, or (in the case of The Children's Crusade) have less of a focus on the Young Avengers. Those other stories also tend to depend on your knowledge of the goings-on of the Marvel Universe at large. Civil War, Secret Invasion, Dark Reign, and House of M (which is related to #6) all are big deals in the Marvel Universe, but if you're not familiar with them, then the related Young Avengers stories will be a bit less easy to get into.
In terms of "Which of these tell an on-going story," the answer is that most of them don't. They're largely dependent on other things happening in the Marvel Universe, and they're published sporadically. Each one will pick up the team where they are at that point in time, which is mostly where you last saw them, but it's not quite the same situation as when you read a proper regularly-published comic. #1 and #7 are the two that are different in that they are more centered on the Young Avengers team.
If you want to follow specific Young Avengers members to other books, I recommend looking them up on ComicVine. Characters have character pages, which in turn have links to all the character's appearances. Here for Kate Bishop/Hawkeye, you can see that after Young Avengers vol 2, she showed up in Matt Fraction's excellent Hawkeye. You can do the same for other members of the team to follow them to wherever they've ended up.
Note, that the Avengers Academy that you mention in your question isn't here. That's because Avengers Academy tells the tale of a different group entirely. During Dark Reign, Norman Osborne finds and experiments on many teenagers with powers. Post-Dark Reign, the Avengers provide them an offer to train them to become future superheroes. If you like Young Avengers, you'll probably like Avengers Academy, just know that it's a different group, and it's telling a different story.