Katie,
My thoughts were the same as those mentioned by Frank. Taking a first trip to Europe in the peak of summer tourist season with no planning is not something I'd recommend.
To answer your questions, you might consider Naxos and Mykonos. Both will probably be busy and crowded, and you may have difficulty even finding a Hotel room. Santorini will probably be packed at this time of year, so I didn't recommend that (it's also NOT "less touristy, less expensive). I was there in June, and the crowds were bad enough then.
You might find it helpful to have a look at Matt Barrett's website as there's a LOT of information there on travel in Greece.
Given the circumstances and the fact that you only have 20 days (which includes travel times), I'd drop Istanbul this time and focus on Greece and Italy. There's more than enough there to keep you busy for 20-days! I'd also suggest increasing your time in Rome. Two days is not enough.
Regarding a "part of the city to stay in" I also like the Termini area, again as it's so convenient for transportation links. However, if you don't already have a Hotel booking, you may have to take what you can get.
You might want to pack along copies of the Greece and Italy Guidebooks, as they'll be a good source of reference if needed.
Happy travels!