Providence sits at the center of a small state, which means most of Rhode Island is within an hour's drive, and southeastern Massachusetts is equally close.
Newport is the obvious first day trip — 30 miles south, accessible by car or bus. The Gilded Age mansions on Bellevue Avenue (The Breakers, Marble House) are genuinely impressive, and the Cliff Walk along the ocean is free. The downtown waterfront has good seafood restaurants and a working harbor.
Roger Williams Park and Zoo is within Providence itself, about 3 miles south of downtown. The zoo covers 40 acres inside a Victorian-era park and holds over 100 species. More manageable than a big-city zoo, and the surrounding park is worth walking on its own. rwpzoo.org
Bristol is a small harbor town 18 miles southeast, best known for claiming the oldest Fourth of July parade in America (since 1785). The Herreshoff Marine Museum covers the history of America's Cup racing, much of which happened here.
Cape Cod is about 90 minutes from Providence, making it reachable for a long day in summer — though it's better as an overnight.