Bob - you will probably do better staying off the island. We don't have an abundance of good campgrounds on the Island. Check the following.
Town of Portsmouth RI:
Look at Newport County (would include island towns of Newport, Middletown, Portsmouth) & South County (South County campgrounds are 15- 40 minutes away and over a bridge with a $6 toll one way for an RV.
Visit Rhode Island :: Where to Stay :: Campgrounds
When you say bikes, do you mean motorcycles, because you will need a rental car if you are using bicycles. Taking your A class into Newport to see the mansons can be a challenge. Especially parking. It's a tourist town but they don't get it, some tourists come in an RV.
- helpful to get a picture of the campgounds.