Ellen, I think you will have to get a Ukrainian visa before you go, unless Oceania takes care of that for you--Radisson did all that. We got proper stamps in our passports, and were required to carry them off the ship. But we were allowed to travel independently. I would contact Oceania and ask if they do the same.

Odessa you certainly can do on your own. All three Ukrainian ports, actually. Just do your homework first, and realize that everything is in Russian and you can't count on English speakers everywhere. If you want to see the Crimean war sites in Sevastopol and Balaklava, you might want a guide.

I think the only place I would hesitate to do without some kind of guidance would be Constanta. It is a big industrial port, quite a big city. Outside is a very large beach resort--lots of Europeans there, seemed quite nice. But Romania in general just seemed a little edgy to me. But that might just be me--there's lots of contradictions, beach resorts, fancy shoes for sale in the stores, snotty teenagers slouching around, and men working in the fields with pitchforks, and villages full of horse-carts and donkeys. The Danube basin is beautiful, full of wildlife, but also scarred from industry.