That's a really general question and a huge area. Do you have anything on you special list - peak, lakes water, desert mesas, rivers, fish, forest, wildlife.
In 10 day you might do a really interesting combination rental canoe down the Green River canyons with side hikes, then backpack out through Canyonlands national park (canoe gets picked up down river). Guaranteed that is nothing like the AT. LOL Might be a bit to warm then.
On the other hand that time frame is perfect for the Sierra, Olympia NP. Montana.
I'm guessing if you get *more* specific you will get more suggestion from folks here.