Hire a bike from one of the rental shops near the pier — there are several, all charging around €12–15 for the day. By bike, Dún Aonghasa is about 45 minutes each way; the road west is flat for most of the distance and only gently uphill on the return. The island is 14 km long and you can see most of the main sites in a full day without rushing.
Horse-drawn jaunting cars operate in summer and give a slower, more leisurely version of the same route. Minibus tours also run from the pier if you prefer not to cycle — the drivers know the history and the commentary is usually good. The island has one main road running east-west; you are unlikely to get lost.