Chagrin Falls is perhaps the most worthwhile day trip from Cleveland. This small town southeast of the city has quaint shops, funky places like Dave's Cosmic Subs, and surprisingly upscale boutiques. Blake's Seafood has excellent window views of the picturesque waterfall. On the way, the Eton Collection shopping center on Chagrin Boulevard has unique stores and restaurants.
Cuyahoga Valley National Park lies between Cleveland and Akron and is certainly worthy of a day. The historic hamlet of Peninsula sits nearby with little shops, restaurants, and a hostel. You do not need a car: the Towpath Trail bike path follows the old Erie Canal route from downtown, and a scenic train makes stops through the valley.
Cedar Point in Sandusky is one of the most popular summer destinations for Clevelanders. This peninsula on Lake Erie has more roller coasters and rides than almost anywhere else in the world, with an excellent reputation for cleanliness and safety. The Lake Erie Islands nearby have campgrounds, wineries, bed and breakfasts, cottage rentals, marinas, and beaches. On the drive out, the historic whitewashed harbor town of Vermilion between Cleveland and Sandusky is worth a stop.