Hi
We stayed at the Gardenya earlier this year. There is a beach in Bodrum, about a 10 mins downhill walk from the hotel, not the best beach I've been on but nice for dining or having a beer next to in the evenings. There are, however, many lovely beaches just a short dolmus ride away from the local bus station. We explored the peninsular and found at least seven lovely beaches to swim in and walk along and have a drink by. Yahsi, Gundogen, Yalikavak, Gumusluk (my favorite), Torba (very close) are all well worth a visit as is Bitez. In my opinion the ones to avoid are Gumbet and Camel beach.
I'm sure you'll have a lovely time!