One bump

I've never ridden there but I have traveled through several times. I would guess the "must-do" (epic) ride is to ride from Durango to Silverton, and possibly back. Major climbing. You can also do it one way and possibly take the train back, though I'm not sure on their policies on bikes and sweaty cyclists.

Highway 550 from Durango to Ouray is one of the most beautiful rides in the US. It is 70+ miles one way and crosses two fairly high mountain passes. As posted above you could stop short in Silverton for an out and back but it would still be a long day.

Do you mountain bike? That is the most fantastic riding around Durango, hands down.

Any leads on where to rent a nice road bike in town?

Sorry, not really.
But Mountain Bike Specialists is a super cool off-road shop in downtown, if they don't have any road bikes (they may or may not I haven't been in a while) they will be able to point you in the right direction.
The guys at that shop were super cool on my last visit, gave me the low down on a bunch of cool trails to check out. I'm sure they will help out a roadie!
Have fun, it's a cool town.