Bakhchysaray is Crimea's best base for trekking, with beautiful mountains surrounding you on all sides. Owners of many hotels listed following will happily assist with planning routes and often accompany their guests on the treks. Maps of trekking routes are available at the Tourist Information Centre. Rangers from Mountain Rescue Service are the most experienced guides in town, but you'll have to devise a way of overcoming language barrier.
If you are into cycling, hook up with the Hotel Koleso owners. They can deliver a mountain bicycle to wherever you are in Bakhchysaray or you can join one of their cycling tours.