10 days may seem like a lot but it can go by rather quickly.
Athens deserves at least 3 days to fully experience the city. That leaves 7 days . . . you could combine two islands but don't forget every time you move it eats up time and time is valuable on holiday.
Santorini & Mykonos are two of the most popular of the Greek Islands and July will be High Season so expect masses of tourists being spewed out by Cruise Ships. They are also two of the most expensive islands. How "Greek" they are depends on what you want for a Greek Island.
There are numerous islands that don't see the crowds but as stated High Season will find many of them full of tourists, some more or less but still busy.
The first time we went to Greece we used Aegean Thesaurus and were more than happy with them. They arranged accommodations, transfers, ferry tickets, etc. We booked our own flight and once we reached our destination we were on our own, but since we did a lot of research we know what we wanted to do.
Since the first time we went we have done our own bookings for everything and save money for more important things . . . Greek Food!
I've never used Air BnB so can't comment on it. You may get a better price but during High Season everything will be more expensive.
We use booking.com or trip advisor to search for accommodations and when we find something that meets our needs we go directly to the accommodations website (if they have one) and deal with them. You'll probably get a better deal and customer service.
Numerous islands have their own websites with lists of accommodations.
Or go to: http://www.gto.gr/index.htm. Click on an island group and you'll get lists of villages with accommodations.
The best website for Greece is Matt's Greek Guide: http://www.greecetravel.com/
Two very good guide books are The Blue Guide to the Aegean Islands and Greek Island Hopping with the Blue Guide the better of the two.
One advice I always give is "Do Your Research". Don't let a travel website or glossy magazines decide where you want to go. There are many Greek Islands and parts of the mainland that don't get the hype like the Big Name Islands. Maybe those less known areas would provide you with a more truer Greek Experience.