Take a look at staying in Dillon/Silverthorne. It's 1 hour, 45 minutes from the Denver airport according to Google Maps and you'll be close to Keystone and Breckenridge ski resorts where you can rent any type of bike you could possibly imagine. Also, since there are a ton of vacation rentals in Dillon/Silverthorne you should be able to get a good deal this time of year.
As far as trails, you can take your pick! From lift serviced to epic backcountry rides, you'll be near it all. I don't know about short rides in the area but I'm sure some of the locals here can fill you in. Any trail can be short - you just have to turn around before you go too far.
I love staying in Boulder myself... once you're there you don't need to get back in your car. Everything is very walkable and bikeable. Beyond the two trails skibum mentioned, there's also Valmont Bike Park, Marshall Mesa, and Heil Ranch nearby.