The Sporades Queen aka the glass bottom boat is already docked at the harbor front after its winter maintenance. There will be others by June. The routine voyage is a trip out to sea Alonissos and the Marine Park.
For restaurants in June there will be plenty along the port (10 - 15?) and easy to find. Don't forget to explore within the village for restaurants as there are some very good ones including Anna's, Rodi, and Alexander.
I really hope that Riginos' Perivoli is open by the time you arrive (he is renovating). It is behind the Platanos Square in from the port. This place is a must.
Also outside of town along the "Ring" road is a quiet place called Peparithos which good food and nice ambiance.
There are also pizza shops here and there as well as the already mentioned souvlakia (gyro) joints.
In Agnondas Litsa mentioned Pavlos Fish taverna. It is very good.
My experience is that you can eat inexpensively if you order more appetizers and salads, cut down on the meat and fish main plates, and order the house wine. The more you eat the more you pay. My family ate the other night at Gialos (along the port) for about 12 Euros each - using the appetizers/salad/house wine formula.
Use Trip Advisor to see the restaurant reviews.