While there is not a ton of things to do in Washington, there is still a decent amount. If you're in the mood for a movie, check out Carmike Sunnyland Cinema 10. They almost always have all recently released movies showing.
Plaza Lanes is a good place for bowling. You can reserve lanes online if you want to make sure that you're able to bowl when you want to. The owners of Plaza Lanes are also working on moving away from only bowling and expanding the place into a full-blown "family center." Plaza Lanes also has a bar and grill.
A great place to get some exercise is Farmdale Reservoir, maintained by the US Army Corps of Engineers. There is about 20 miles of trails here some or all of which are good for hiking, biking, horseback-riding, and cross-country skiing seasonal.
You can also be active outside by making use of the Washington Recreational Trail. This walking, jogging, and biking path runs for about 3-4 miles from N. Cummings Lane to N. Main Street.
Golfers will be glad to know that Washington is home to three golf courses. These are Hillcrest Golf Center Inc. located on Washington Road; also has a small miniature golf course, Quail Meadows Golf Course and Driving Range located on Centennial Drive; includes a pro shop and indoor golf course, and Pine Lakes Golf Club located south of Washington on Schuck Road.