The best bike route from the Old West End to UT depends on the time of day. During the day, take Delaware past Detroit to the (pretty rough) bike path, which will eventually take you to Bancroft a few blocks away from the University.
PROBLEMS WITH THIS BIKE ROUTE: There is no red light in sight when you cross Monroe at South Cove. The more direct bike path around Ottawa Park is under construction. This route intentionally avoids the light at Parkwood and Bancroft, because it is a stop light controlled by a scale that will not be triggered by a bicyclist.
At night or other times when there is less traffic, ride on Bancroft.