Day trips are touted in all of the tourist hubs. One of the most popular is a day trip to Ephesus. It’s a 3-hour journey in each direction, and some excursions will get you there and back in a day, giving you 1/2 day to tour this ancient Greek site.
If you want more flexibility when you visit Ephesus, you can rent a car, and stay overnight in the nearby town of Selcuk, and then spend a leisurely morning touring Ephesus.
If you’d prefer to tour ancient ruins a little bit closer to the Bodrum Peninsula, on the other side of the Milas-Bodrum airport is the ancient city of Iassos, and in the hills above Bodrum, is the ancient city of Pedesa.
While neither of these is in the same league as Ephesus, they’ll get you your ancient-site fix. You can hike up to Pedesa from Bodrum, but you’ll need a car for Iassos – and that trip will take you about 90 minutes from Bodrum.