Hi-
The bike shop you rent from should have a good map to give you of bike routes.
Some suggestions:
One of the most popular routes is called the Lake Monona loop. It is signed fairly well (except where it crosses Monona drive). It is about 12 miles and a combination of some trails and some lower traffic city streets. North of the lake the signage directs you from the bike path back to city streets, but you can easily stay on the bike path.
Another loop is to take the Capitol City south of Lake Monona, through the nine springs area and Fitchburg where it connects to the Southwest Commuter trail to bring you back downtown. That is about 17 miles. That loop has a small overlap with the Lake Monona loop so you can combine the two to get close to your desired mileage.
I enjoy out and back biking on the Military Ridge trail, which is a bit south of Madison but reachable by the Capitol City/Southwest Commuter loop. However, the lovely part of it out in the country is packed dirt, which I will ride on my road bike, but I'm used to it!
Keep in mind that you need a pass for many of the trails, which you can purchase at many bike shops or kiosks along the trail.
As for rentals, I did a quick search on Yelp and it gives some options. I obviously own mine, so can't comment much there other to say that it seems like most shops provide online pricing so you can compare equipment and cost in advance.
Keep in mind that weather in October can go either way- sunny and warm or not so nice!