The hills of Umbria make cycling not a sport for all. However there are also routes you can do that avoid the hills. For some very well documented cycling routes you can go to the website www.bikeinumbria.it.
Renting a bike can be done at Mate Bicycles. They have a good choice of hybrid bikes and racing bikes. Mate is close to Magione (just north of Perugia). Mate can bring the bikes to your apartment (not included in the rental price). If you would like this, do make sure you have clearly indicated your height and any special requests you have. Best send them an email so they can reply in English with the price also for the transportation of the bicycle. It is difficult to get a bike last minute in high season. Make sure you book this in advance!
In Foligno is Ecologico Tours: Ecologico tours bike rental. They also have E-bikes for rent at very acceptable prices and can deliver to your holiday home. Make sure to book in advance! They speak English.
In Castiglione del Lago one can rent bikes and make a nice tour around the Lake: Cicli Valentini.
Also in Passignano sul Trasimeno there is a bike rental, the Lake Bike Store.
In southern Umbria Randotrek may be a good contact, they deliver the bike(s) at extra cost in Southern Umbria.
Umbria & Bike promotes holidays by bike in Umbria.
Very popular is Umbria in Vespa, based at Lake Trasimeno, near San Savino and the Oasi La Valle. The owner is German and she speaks English fluently.