STanbr and I are a "Tag Team" -- between us we cover the usual Cyclades Island itineraries (that's the group you are looking for). I agree that booking.com is your best way to go, because you can see availability instantly ... and please, do not dither on choices because ALL islands are filling up fast fast fast for your period. Here are 2 further thoughts:
• Do NOT attempt to get an accommodation that holds all 6 of you --- MUCH harder to find, in the locations you will want. Much better to get rooms adjoining/nearby. (and besides, every pair gets their own bathroom, and peace & quiet if they want to nap!). If you are a sizable guy & at home use a queen/king bed, you'll have much more choice of lodgings if you are OK with twin-bedded rooms; fewer Greek lodgings have queen/king except modern & luxe places, and you've already indicated budget concerns. BTW, when u ask advice re lodgings, it helps advisers if you state your preferred budget range per room per night, in Euros (http://www.xe.com/ucc/ = instant conversion site). Santorini will be most expensive, then Athens, then Naxos.
• When viewing properties, use Google Map on page! It shows where hotel REALLY is, not what text claims - So often they say "near acropolis" or "on Beach" or "near Caldera" when they're not.
SANTORINI - if you want a Caldera view, make sure this is stated; when a site says "sea view" it usually means to the East, not the sunset side. Do NOT choose hotel on East side of the isle, since the Famous View is the main reason to go to Santorini. It will be costly, but compensate by a shorter stay (3 days?). And if you can't find, or can't afford, rooms with a view, sometimes there's a roof terrace where all can gather. On west main choices are Oia,Fira town, Imervogli, Firostephani. Oia is super-costly, Fira is super-crowded, Imervogli is quiet, but usually requires car rental (and only 5 can legally fit in a car in Greece). Firostephani, IMHO is the "goldilocks choice" -- many many hotels w. caldera views with the Plus of being an easy walk into Fira Town & back, for shopping, museums, nightlife. A pool hotel? (and it's HOT in July) alas, prepare to pay $$$. In Oia, there's a Hotel Anemomilos?? where nonresidents can use pool for a fee or buying lunch there.
• AGENCIES vs DIY - Agencies are more expensive, that's right. After all, they have to make a living, and they'll recommend hotels that pay commissions. Also, their job is doing the work FOR you. It's the old story; either spend money, or time. A wonderful trip D-I-Y is very possible, it just means putting in some hours, and making some decisions ASAP, due to your timing for High Season. Good luck! Get transport figured first and as stanbr says, this forum can comment on your hotel list.
• HAPPY PS -- re Mykonos - you actually can have your cake & eat it too, if you stay on Naxos. There's an excursion company there that runs 2 day-trips per week; you spend 3 hours on Delos, a few hours in Mykonos town or at the beach, and you're back in Naxos by dinnertime.