Newport's famous mansions are about 30 minutes from Shepherds Run. There is a very cool 3.5 mile cliff walk that you can do on your own between the sea and the mansions. There are also self-guided audio tours around the mansions and some tours inside the mansions themselves. If you are hoping to go inside the mansions, tickets are usually required to be booked in advance.
While it will be too cold to get into the water or lounge on a towel, walking along New England beaches in October can be quite lovely! East Matunuck State Beach, East Beach, Captain Roger Wheeler State Beach, Narragansett Town Beach are all good options nearby. Since it is off season, it shouldn't be too hard to find parking and shouldn't be too crowded.
If you are looking for hiking, there are a range of options. Beavertail State Park is nearby and has really beautiful views, but the trails are relatively short and flat. If you are looking for a bit more of a hike, the long pond woods trail near Hope Valley, Rhode Island offers some great lakeviews with 500 feet of elevation gain and is only about 30-40 minutes away.
If you are looking for more city-style activities, there are some great options in Providence. The State Capital at surrounding grounds are beautiful. The RISD art museum has some wonderful exhibits. You can also explore Brown's campus, wander around various neighborhoods, or grab a bite to eat in the area.
If one of your top priorities in coming to Rhode Island is learning more about Taylor Swift's song 'Last Great American Dynasty,' you can check out her beach house at Watch Hill. While you probably won't get past her security, the rest of the town offers a beach walk, fun restaurants, and a perfectly manicured lawn to play croquet at the Ocean House.