It really doesn't matter where you go in CO, it will be great! No joke. I'd say pick a place you want to visit and go. EVERY mountain town has great mountain bike rides. Summit County (Breckenridge, Keystone, Copper Mtn) has more trails than you could ride in a month, with excellent maps available. At Vail, you can take the gondola up with your bike and ride down. Expensive day ($35?), but fun nonetheless. Aspen, Crested Butte, Telluride: priceless.
The only caveat about June in CO is snow. While most snow should be gone in June below 8-9000 feet, it will likely still be impassable above that. After all, June is typically our runoff month when the rivers are at their highest. It really depends on what kind of winter we have. The temps will be nice though. This past summer it was generally 67-70 in Breckenridge. Perfect for any activity.
I'm not too familiar with the southwest part of the state, so I would get some opinions on whether the snow will be gone by then down there.
The pic below is me hiking somewhere around June 10, 2001 after a drought winter. The pic is about 11,500, which is much higher than you will be riding. But, it does show you some interesting things. First, lots of snow. Second, it was warm, at least until the clouds came in!