12-day itinerary for family of four, twin daughters 22

Does 12 days include travel to/from Greece? You cannot count the day you leave N America (I'm assuming you are from US), OR the day you arrive in Greece unless land before noon, then a half-day. You can't count departure day at all. So... it might be 8 1/2 days. A "few days in Athens" shrinks island time down to as little as 4 1/2 days -- at most, 5 plus. Thus you may have to consider just ONE island (& maybe daytrip to another), or your trip will be a blur, a drive-by.

And no June is not "okay for reservations" -- given the world-wide "discovery of Greece" June is almost high-season these days, especially for transport. With short trip, you can't spend 6+ hours on a ferry ... and domestic flights may already be full for some isles. And before yo u book lodgings, you need to know how you'll get there & back. One must be in Athens vicinity the night before departure flight -- too risky to returnto ATH area on same day; any glitch means you miss flight & pay $$$$ for one-way tix. The most time-efficient way to do this is Island first, Athens last.

For us to give any useful advice, we'd need specifics asap -- What are exact dates of Flight Arrival/departure (AND exact time of day -- every hour is precious for options). It's useful to look at a map & know something about Island connections, Greece is not like Itally or France, where massive frequent trains make connections fast & easy. An agency would handle all that, but if you are DIY, you'll need to do some homework. So, for islands of interest, you need to know if flights are still available on your arrival day, and for the day you plan to return to Athens. You can easily check this on the 2 domestic airlines, Aegean/Olympic -- http://en.aegeanair.com/e-services/e-ticket/ and Sky Express - http://www.skyexpress.gr/

IF fllights permit, I would suggest on arrival (allow 2 hrs layover), try to FLY to NAXOS, spend 4 days minimum, then (maybe) 1 hr ferry to PAROS for 2 nights, fly back to ATH (OR take 7pm ferry arrive Pireaus midnight). Before you mourn Milos, read more about how it's changed.

As for accommodations (AFTER you lock in transport), it's all about location, location ... and you'll do best if not looking for 2-BR apts on islands ... for family-owned places on or near beaches, late-bookers do best if you are Ok with 2 separate BRs/units. After all these are grownup young women. In Athens, booking dot com now offers a lot of pvt apts on short-term, in good locations, at rates as good or better then hotels. Id prefer that agency to the vbro or airbnb approach because better guarantees in case of gllitches.

Janet has already told you everything.

Milos sounds incredible.

While I don’t want to dissuade you from visiting Milos, be careful not to fall into the trap of Instagram trends, especially with Milos and its famous volcanic beach of Sarakiniko, and the equally photogenic "syrmatas" of Klima, the typical fishermen's houses (almost all converted into Airbnbs). If renting a car isn’t an option for you, exploring Milos could be a challenge. The bus service is limited, as the island mostly has just a couple of villages: Adamas, the capital and port, and Pollonia to the north..

There are a few beaches in Pollonia, good accommodations option and quite a few places to eat, drink, etc, but all the other beaches are difficult to access without a car. Keep in mind that a large part of Milos is uninhabited and designated as a Natura 2000 area (the zones marked “No Service” on the map below, requiring a 4WD rental).

https://milosrentalcar.com/rental-terms/

Paros or Naxos would probably be better suited to what you're looking for.

I can't comment on Ios, as I've never been there. It has always been very popular with young people who don't want to break the bank like in Mykonos. As with Milos, I don't want to discourage you from going to Ios either, but note that getting to Ios takes 4.5 hours by ferry if you take a fast ferry early in the morning. Leaving later, it takes between 5.5 and 9 hours, arriving at night depending on the type of ferry.

Although Paros and Naxos have a decent bus network (the bus routes on Paros are better planned than those on Naxos for getting around the island).

Also, think twice before saying you won't have a car. It will be the best way to discover the island you're staying on. It would be a shame to fly thousands of miles only to get stuck in a small town on a Greek island without seeing what other parts of the island have to offer.

Whatever the island, as Janet pointed out, don't focus solely on "hotels." There are many more family-style accommodations available, whether rooms, studios, apartments, or even entire houses. The services and comfort are just as good, and often even better.

We've booked Airbnbs on the islands quite often (almost since Airbnb started) without ever encountering any problems.
We only book Airbnbs run by individuals, never by professionals. VRBO isn’t as well-known in Greece and Europe compared to Airbnb.