The last time we were there we ate in the Eleni on the main street
Tavern Hellen (Ελενη)
2731 302524
https://goo.gl/maps/Ac61ogeBGEDU11yw6
The street is quiet at night and it’s atmospheric rather than pretty. There were a lot of locals eating there, the people were friendly and the food was good. If you don’t fancy it there are two or three other places within a couple of minutes walk.
I would certainly opt for a dip afterwards. We were there in September and it was hot. June will be even more so.
It’s a big, and very steep, site so two visits might be a good idea. It would also mean avoiding being there in the middle of the day. There are two entrances: one at the bottom of the site and one half way up. If you make two visits I’d do the top half of the site in one and the bottom in the other. If you decide to do it in one visit I’d take a taxi to the upper gate. It will only be a few euros and it will save a long walk up the site at the beginning (if you drive yourself to the lower gate) or, worse, back up to the car at the end (if you’ve left it at the higher).
Alan