I excavated here in July and August 2004, so my information might not be really up to date. It is a weird mix of touristy beach, archaeology and religion. The oldest church of the Ukraine, or at least the place of the first church in the Ukraine, is inside the Chersones reserve. It is the first thing you come upon when entering the site. It is definitly worth a visit and not just for the building itself. It is still actively used as a church and it has a very particular atmosphere. Make sure you have appropiate clothing.
The archaeology is a bit spread out over the place and there is not much explanation to be found about it. I suggest you visit the two museums first and then if you have some money to spare buy a guidebook. When I was there there was only on book to be had in English all the others were in Russian or Ukrainian.
The beach is a pebble beach and there are a few places were you can buy some food and drink but most people brought their own stuff. There was a little bar on top of the hill between the church and the main archaeological site were you could have a beer or an ice-cream and watch the sun set over the Black Sea. Well worth staying for.
There was a small entry fee to enter the reserve, don't remember how much is was but it wasn't much.

Muzeum założone przez archeologów z Polski, na terenie wykopalisk greckiego miasta, założonego ok. 2,5 tysiaca lat temu... Warto zwiedzić :)