Well the season will have just started in May so everything will be up and running. The weather will be warming up nicely, although you may need a cardi for the evenings. Turkey caters well for vegetarians so you won't have any problem. If you don't see anything on the menu that suits you, have a word and they'll be able to come up with something for you. Turkey uses an abundance of fresh vegetables and they are really tasty. The mezes may suit you - a dish containing a selection of tasty small dishes, just tell them what you don't eat.
The beach in Bitez is lovely and you'll be spoilt for choice with the cushions, hammocks, etc on offer. Each restaurant along the front has its own bit of beach which you are free to us as long as you buy a drink from them.
As for trips, you can always try a boat trip - a lovely chilling day, choose a hot day and you will be cooled by a lovely breeze. Take your time in booking - if you walk along the front at Bitez, I know there's a boat trip advertised which is really relaxing and not packed but sorry I can't remember its name. I'm sure you'll see it though. Take a look at the different trip agencies and see what's on offer - I'm not sure what your interests are but there will definitely be something to suit you.
Also explore the peninsula under your own steam. Get the dolmus into Bodrum and from the large bus station in Bodrum, you can get another dolmus to other places on the peninsula - Yalikavak, Turgetreis, Gundogan, Gumusluk, Gumbet and not forgetting Bodrum itself. The castle in Bodrum is great to look round, even if you're not into history or artefacts, the views from the castle walls are great.
Hope you have a great holiday - we will be there just before you as we fly out on 25 April and we're staying for 10 weeks. Our house is on the Gumbet/Bitez border and we love the area. As well as the restaurants along the front at Bitez, try those set a little further back. If you need to know anything more, just ask.
JAK xx