Since I'm more into the old shit, I'd recommend staying in the mediterranean, but there are a lot of sweet places regardless. What I would do is pick a start and end destination, book the flights, then fill in the major moves within the itinerary. The summer can be pricey, but you can save a lot by making some good decisions.
First things. The euro can be fucking brutal. It isn't too, too bad right now, but keep in mind that peak season just adds major costs. Countries like britain, switzerland, austria, all of scandinavia, are pretty wicked expensive. I would probably avoid those to save some coin.
If you want to visit a couple of counties that are far enough away (ie italy and greece, italy and spain, etc) I would opt for flying rather than the train. This is especially true if you make the choice way ahead of time, then flights like on easyjet can be dirt cheap and you save tons of time. If you want more spontaneity, which in my experience has been considerably more expensive, a euroail pass is good to have for the expensive trains, but keep in mind you need to ride a few wicked expensive bullet trains to make this worth it. I like to travel where I have major moves planned out to save some money, then fill in regional shit as i please when I'm there, For instance, I will just buy train tickets, get a room or something while I'm in... umbria, but I will have bought plane tickets from lyon to rome a few months in advance. Gives flexibility, but saves cash money for things like better food, drink, party, etc.
As for places to go, I'd probably just hit a couple countries max and actually spend some time in the places. More enjoyable than just moving around for 2 weeks. You mentioned that the Greek islands were of interest, and I would still recommend them. Athens is great for a day. Greece is nice because it is on the Euro, but it isn't as inflated as other Euro countries. You can find some really cheap places to stay, even at the most expensive destinations (Fira on Santorini comes to mind) with little to no inconvenience. It would be nice, however, to stray a bit from the main tourist islands. You can hit up like paros or santorini or something, party through the night and morning, then hit up like falegandros or kalymnos (requires some finess with the ferry schedules greekferries.gr) and let time slow down a bit. Beaches, sun, old world, sick. If you want some of an interesting mix of all shit awesome. Ultimately, island hopping is sick. I'd also recommend Turkey. There are some beaches if you're interested, but really it is badass wherever you go. Easily linked up with Greece, by air or by boat (samos to kusadasi, kos to bodrum, rhodes to mamaris, etc) Probably among the best mix of the ancient available, and clash of byzantium and ottoman. Istanbul is one hell of a city, where you can hit up the coolest shit (hagia sophia is easily one of the most impressive buildings on earth, especially considering its long and dense context), party hardy, and save a shitload of money. two weeks in greece and turkey could be fucking insane. Add in a little more time and you could start up in rome with a 2 hour flight to athens, as a for instance)
You could stretch money far as well by going to countries like Croatia, Hungary, Czech Rep, which are all fantastic places too. I just went on a tangent with greece and turkey.
