Asheville resident here. For hikes check out Dupont State Forest or head up through Brevard and check out a lot of the trail heads that start outside of town (Looking Glass is a good one). Panthertown Valley is a sorta of hidden hiking gem as well. The grounds at Biltmore are gorgeous and like everyone else said if you are fan of history definitely check it out. Grove Park is pretty cool too (and free to get into).
This is the land of breweries. Some recommendations are:
Greenman.. They just tapped a bunch of cellared beer in celebration of their 20th anniversary. They've got verticals of their holiday Holly King and their stout Dweller (and bourbon aged Demon Dweller) on right now. Both are high ABV so beware. And both are awesome.
Burial: Just damn good beer, all the time. They do a lot of saisons which is a style I don't like but anything they do is going to be good.
Funkatorium: If you like sour beer check this place out. It's an off-shoot of Wicked Weed (the brewery that just sold out to In-Bev). The main Wicked Weed brew pub is a tourist trap but the Funk is a cool spot to chill and get a good sour in.
Those 3 are probably my favorite. You can get New Belgium and Seirra Nevada stuff anywhere so I tend not to hit those places up too often. That said, both put a lot of work into their space and if you have the time definitely check it out. Plenty of other breweries as well. Hi-Wire, Twin Leaf, Catawba, AVL Brewing company, One World (cool little hangout place), Lexington Ave Brewing (meh), Bhramari to just name a few. And they are all pretty much within walking distance of each other.
Lots of good music venues here. Jump online and see if anyone interesting is playing at Salvage Station, Grey Eagle, The Orange Peel, AVL Music Hall, etc
Edit: It's kinda cheesy but consider doing the La Zoom bus tour. It's pretty damn amusing and it'll give you a good layout of the town, what breweries you might want to hit up, etc.
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