Now that Mouse has broached the subject, I'd gladly put a plug in for staying in the East. I think our backcountry might be less intimidating for a beginner, quite honestly. And, if your parents are willing, it could simplify your logistics issues and give them a bit more peace of mind.
If you want a month on the trail, then you could do the first leg of the Appalachian Trail (and finish section-hiking the whole trail by the time you finish college.) Alternatively, if you don't mind breaking your trip into disconnected sections, you could sample a great number of different areas: The Smokies, the White Montains, Shenandoah (all three contain AT segments); you could also visit some of the "pocket" wilds, such as Mammoth Cave, the Hundred Mile Trail (joins the AT in New England), and Isle Royale NP (in the middle of Lake Superior.)
For a first trip, these will be plenty intimidating without being overwhelming. You'll gain great experience, and then be ready to take a big bite out of Big Mountains in the West next year.
I'm not trying to rain on your parade, but I think Mouse has given you a realistic and achievable alternative for a first trip.