Visit some of the other Cinque Ports, or the Hythe and Dymchurch Miniature Railway. To the west Hastings, Eastborne and Brighton are reachable by direct train.
Near Rye passes the National Cycle Network, and you can hire a pushbike from "Rye Hire" near the rail station (friendly service, bikes in good condition, and come with a lock). Sadly the designated cycle paths are not always well signposted, or consist of paths with quite large rubble; you will have to show some determination. But it's fairly flat ground around Rye (although town centre is on a small hill), so for the non-driver this is an alternative to buses.
Ten minutes drive or bicycle from Rye is the beach town of Camber Sands (see discovercambersands.com) which has the longest sandy beaches on the south coast of England. It becomes very popular in the summer months when English people take to the beaches in search of sun.