There are several lactation rooms available on campus. For details, see https://www.ohio.edu/womenscenter/resources/Lactation-Rooms.cfm
Athens City Pool--a brand new pool is set to open any day now. Visit the website to confirm before visiting.
Putt People First Miniature Golf (behind Richland Avenue Park/across from OU Inn; call 740-593-3900 for hours)
The best playgrounds are at the Athens Community Center (701 E State Street), Richland Avenue Park (147 E Ridge Dr and across the street from the OU Inn).
The Ridges--hike, play, and explore the haunted Ridges
A Bit Further Afield...
Outdoors: The surrounding area has many great options for hiking and exploring. Hocking Hills State Park, Stroud's Run State Park, Zaleski State Forest, and Lake Snowden (which is about to open an inflatable water park) are all 15-30 minutes from Athens.
Ohio History: Take a ride on the Hocking Valley Scenic Railway. Visit famous Civil War general William Tecumseh Sherman's home in Lancaster or Ohio's oldest settlement in Marietta, where you can also ride a paddleboat. The National Underground Railroad Freedom Center is in Cincinnati, where the Art Museum is currently hosting a Xi'an Warrior exhibition.
Children's Museums: COSI, Ohio's Center of Science and Industry is an excellent children's museum that just opened a dinosaur exhibit featuring dinosaurs from the Smithsonian. Lancaster's AHA! is a smaller, but closer, children's museum and indoor play space.
Zoos: The Columbus Zoo and Aquarium is a world-class zoo with a number of outstanding exhibits, including the recently opened Heart of Africa. Unless you grew up in San Diego, the Columbus Zoo is well worth the trip. For a safari-esque experience, try The Wilds.