Jebejava is right— the EBP is a great place for casual rides. You can pick it up right over the river in East Providence. There's a ramp over the river by the 195 bridge by India Point Park. It can get pretty crowded on weekend afternoons in the summer though.
Another great option is the Blackstone River Bikeway, which takes you up from Pawtucket to Woonsocket. It runs along the river for a good stretch, passes some nice small falls and old mills. It's a really beautiful pathway.
The Washington Secondary, which you can pick up off of Elmwood / Reservoir in South Providence takes you all the way down into Coventry. You can easily hit the CT state line off of that.
The Blackstone and Washington are both great to include as part of half-century length rides. There's a bounty of nice country roads out in the north and west of the state. As jebejava mentioned, the NBW rides (as well as their website) are a good way to pick up some routes.
We'll probably be doing episodes on Bike Talk focused on each of the main bike paths in the state soon, with reviews, stories, and field recordings. Keep your eyes on bsrlive.com and ears on 88.1 to find out when!