Road or mountain??
For road rides you can ride right out of town south on 550, up Red Mountain Pass. It's a narrow road with big drop offs, but I rode it on a loaded touring bike and didn't have any traffic problems. People are driving pretty slow anyways.
The other way out of Ouray towards Ridgway, not much of a shoulder, fast traffic.
A nice ride out of Ridgway is out to Dallas Divide, and back to Ridgway. I've done that ride a few times.
For mountain biking there are a lot of jeep trails out of Ouray, but I have not mtn bikes them.
Are you coming through Grand Junction? There is a lot of awesome riding around here.