You are lucky.......Telluride is one of the best places in the world.
There are a number of great hikes right from town. The most popular is probably Bear Creek Canyon - an easy two mile hike up a beautiful valley to Bear Creek Falls. There are other trails that connect to this one, and you can go beyond the falls if you like, and up over the ridge into the Ophir valley. .
Another wonderful hike is the Jud Weibe trail on the opposite side of town - this is also short, and amazingly beautiful (waterfalls, wildflowers, views). JW also connects to several other great trails, including Sneffels Highline. Classic T-ride hike.
At the end of the valley is Bridal Veil Falls - the longest falls in CO. You can hike to the base, or up the swithcbacks to the top. The valley above the falls is spectacular. You can see Blue Lake and loop this around over to the Bear Creek Basin if you like. Just get a good map and be prepared if you decide on the loop.
You can walk along the San Miguel River. It is super easy, but the river is beautiful and peaceful. Watch for wild trout.
Tomboy Road is great. You can hike up and over Imogene Pass to Ouray if you are feeling frisky. Amazingly beautiful up there.
If you have a car, consider Alta Lakes (a few miles down valley). Amazing.
Hike the ski area. Take the free gondi to Mountain Village, and then hike away. Gold Hill has stunning vistas if you decide to go for the summit.
There is so much more. Go see John Duncan at Telluride Outside/The Telluride Angler. He can set you up with maps...etc. He is a great resource and just about the nicest guy in town.