Colorado rivers are flooding, but nothing like east of us. Some of our mountains have 300% of normal snowpack, which is melting rapidly due to high temps this week, The Glenwood Canyon bike path is under colorado river water. The Yampa river in Steamboat is out of its banks. However, my guess is that the Rio Grande trail from Glenwood to Aspen is OK.
Denver is fine for bicycling. Here is a somewhat outdated (doesn't show the lates additions, etc.) map of the major Denver area trails.
I would suggest any of the trails. Chatfield State Park, Bear Creek Trail, Cherry Creek Trail and State Park, the Platte River trails - all are fine I personally don't enjoy the C470 trail, but that is because I am spoiled. It follows C and E 470 for many miles and most has just been resurfaced. The Highline Canal is hardpack fines, more leisurely and quite beautiful in areas. It goes 67 miles.
I would stay off the Ralston Creek Trail (slow and disjointed) and the Sand Creek trail - some ugly parts, not really finished.