Hi Russ, welcome to the site. A lot is going to depend on where you intend to fish around Livingston - if it's the Yellowstone River, be aware that it is BIG and fast flowing, with BIG trout in it, so you'll need something like a 9' #7. Not an ideal river for wading, but you can do it at the back of the High School and at some boat put-ins like Carter's Bridge or Mallard's Rest upstream of the town. Most people drift the Yellowstone from a boat but that is an expensive way to fish unless you can persuade your parents to pay the $450 per day rental
There are some nice creeks which flow in to the main river further up Paradise Valley where you'll get away with an 8' #4. The best of these is probably Mill Creek which has some nice cutthroats but note that there is some private water up there so watch out for posted areas (if the fence posts are painted orange, don't go there!) Snowbank Campground is way upstream on Mill where you'll find lots of small brookies and cutts and plenty of public access.
If you're on a budget, keep away from the spring creeks south of Livingston, - great fishing for big fish but $100 per day and there is plenty of free fishing elsewhere
You might also want to try the Boulder River, about an hour east of Livingston, or the Shields River, half an hour NE
When you get to Livingston, go into any of the flyshops and ask the guys there where's fishing best. Dan Bailey's is probably the best known shop and is really great for a wide range of fly-tying materials and tools. Also George Anderson's Yellowstone Angler is a good shop, couple of miles south of town on Hwy 89, helpful staff in there.