About 5 years ago I rode eastbound over the Jamestown bridge then on to the toll booth area of the Newport bridge. After about 10 minutes of hanging out with my bike, a pick-up truck stopped to ask if I needed a ride. I paid his toll and he dropped me off at the first exit in Newport.
There isn't a legal way to ride across the two bridges to get to Newport. The first bridge, the Jamestown bridge, was built to accommodate bicycles but the state has banned bicycles and probably will forever. The second bridge is the Newport Bridge and it is not only unsafe to ride on but they will reel you in.
Here are some options on how to get across:
1)There is bus service, but it only runs a few times a day, handles only two bikes and doesn't run on Sundays: From the RIPTA web site look up route 64
2) As mentioned by the earlier poster you can simply hitch hike across. I have done this. Whether or not hitch hiking is legal in RI I don't know.
3) There is a seasonal ferry between Newport and Jamestown, which avoids the Newport but not the need to cross the Jamestown bridge. This is a very scenic ferry, but only only bypasses the Newport bridge.
4) So you can simply ride across the Jamestown bridge to take the ferry(from May to Sept???), riding the Jamestown bridge is easy to do ( it was legal for many years) and many riders do this, but you run the risk of arrest or harassment by police. Also, I don't think you have much recourse if any thing bad happened.
5) Try a land route north through Providence. Maybe 60 miles? There is no mapped or signed route to guide this way, but there are two bike paths leading to and fro the city but again no easy way to connect the dots. The ride from Providence to Newport is really nice with a only one bad part: a short section of "take the lane" riding. (There is also a Newport to Providence ferry, that I believe is seasonal)
PS: Don't know if you are going east from Newport, but there is another forbidden bridge. The Sakonnet Bridge. Again it was legal riding for years but now it isn't.
You could try heading for Providence and taking the East Bay Path to the end in Bristol.
From there take RT114 towards Newport. You can ride over the Mt Hope bridge on the road but the
barriers are low and its a long way down to the bay. Rt. 114 will take you into town. There is an on road marked path that goes off 114 near the naval station. I cant remember the name of the path