Hey thomdals, welcome to the forum... and welcome to the USA.
Larry gave you great advice in contacting a local fly shop, and joining a TU chapter.
Trout Unlimited is a US non-profit org that protects streams and trout habitat, but is also heavily oriented to fly fishing. Local chapters are spread across the US and have monthly meetings that are pretty informative on techniques, tactics for local streams, and where to go info. Most chapters have group trips to local water, casting clinics, beginning fly tying classes etc, and it's a great way to meet some new fishing friends, and get the lay of the land in a new area. Here's a link for search:
Council/Chapter Search | Trout Unlimited - Conserving coldwater fisheries
The university may also have a student fly fishing club, so you should check with the Student Activities Center. You'll likely run into other students that fly fish too even if they don't have an organized club-- especially if you drag your rod out for some practice casting on the lawn.
As far as trout streams in the area, some of the more popular ones near Burlington are Otter Creek, Lewis Creek (it also gets a run of steelhead from Lake Champlain) and the Winooski. Here's a link with some area streams and updated reports.
River report for Vermont fly fishing sites
There are also a lot of other streams like the Battenkill, Mettawee, White in VT, and streams in NY state like the Ausable and other streams in the Adirondacks that would be excellent to explore, but are a bit further away.
Good luck, and let us know how you're doing. You should have some great fishing in VT.