I don't know about renting bikes in Reno, but Bicycle Bananas in north Reno is a great resource.

Here's a link for some road rides: http://www.pedalers.org/. Go to the "route slips" link. If you're into suffering, there's the ride up Geiger Grade to Virginia City or up the road to Mt. Rose.

There are a lot of mountain bike trails coming out of Rancho San Raphael park. You can also hike those trails.

We still have a TON of snow up in the mountains, so I don't think any of the Tahoe trails will be open. It's a crapshoot whether or not they'll be open in July .