Newport is on the island of Middletown and islands, even sizable ones, can limit your riding options. 30+ Years ago I managed a bike shop on Thames Street called 10 Speed Spokes. We did a brisk bike rental business. Don't know if that shop is still there but I'm sure a good Google search would turn up a couple of rental shops.
As far as the riding, it's a tourist town, full of tourist traffic. A loop around Ocean Drive, about 10 stunningly gorgeous miles, combined with an ocean swim is a must do. For longer rides you can piece together a ride that snakes up through the middle of Middletown (pun intended) and see some horse farms and more rural scenery but some of the main roads are busy, narrow and filled with tourists who don't know what they're doing. The better ride might be to get over the Newport Bridge- I haven't biked over it in years so, check on it's bike access but cross over to the island of Jamestown and loop around over there. MUCH quieter.