I've been to greece 13 times since 1999, the most recent in May 2019... and have planned literally hundreds of trips for newcomers. You have received some good advice on this thread (which I can't refer to unless I open another browser window, because when one hits 'reply" all previous postings disappear ... grrrr). However, I can give you the basic 10-day recommendation I have for newbies ... especially going in high season.
• LOGISTICS are all-important, and not over-scheduling. Remember it always takes longer to transfer from 1 place to another -- minimum of a half-day. So your max would be 2 short-stop islands, one longer, plus 2-3 days in Athens... and you'll need to fly mostly, just 1-2 ferries.
• TIMING - July is worst timing - hottest month, biggest crowds, highest prices. Unless the adults are required to work 5-day week, It would be smart to fly on WED night or THURSDAY... thus arriving next AM on a Weekday morning not Saturday, which will be overcrowded. It will also get you a (somewhat) cheaper domestic flight. Example itinerary for Thursday July 8 thru Sun July 18 AM.
• JULY 8 - fly CHI to either NY, Newark, PHL or Montreal ... for the only 4 nonstop flights (Delta, UAR, AA and CanadaAir) to Greece. Reason? Shortest trip (9-10 hrs in air); also they all land next day between 9 AM - 10:30AM... this allows you to get a noonish flight to y our first island. IF you take a flight with requires a change in Europe, y ou'll encounter delays, and none will reach Athens befor mid-afternoon even if on time, & hard to get to your first island until dark. When your trip is SO short, ever hour counts. BTW, all the airfares are much the same & since u are in Chi, Canada Air is the best of the 4 economy choices.
DAY 1 (JUly 9) On Arrival, hustle to pick up bags get Euros at ATM & go to Domestic check-in (aegean is best. ISLAND #1 - SANTORINI. Why? Because it has the MOST flights, and is a ferrry "switchboard"... ferries from here to almost all other Cycladic Isles. I recommend staying on Caldera Side (best location - Firostephani). Enjoy sunset at resstaurant in Firat Town.
DAY 2 - Early AM to Oia before tour crowds arrive, then vist rest of Island. Don't bother with beaches (black grit)
DAY 3 (july 11)- Morning ferry to yr 2nd island choice -- NAXOS is ideal.. SO mch to do/see, not mobbed because no cruise ships or jumbo-jets for package tours. Hike, swim, drive thru mountains, visit ancient ruins eetc. its worth 4 days. I recommend staying at Plaka Beach, using frequent local busses to explore port town. Rent a car 1-2 days.
DAY ? if you MUST see another Island, ferry Wed AM to ANTIPAROS. it's ideal, in high season can't get overcrowded... and it's just offshore from Paros, so you explore its port-town of Parikia, then back to peaceful beach on ANTi-P for sleeps. But maybe u wold like to do a day-sailboat trip to ANtiparos, & get back to Naxos for a final sunset.
DAY 7 (Jly 15) - from NAXOS or PAROS airport fly AM to ATHENS, (45 mins); taxi to Plaka area flat-rate around €40. Stay in this hisstoric area, away from big streets, you'll walk everywhere. (Hotel Hera, Phillipos etc is very handy on S side of Acropolis. In late afternoon, tour crowds are gone from Acropolis, light is golden, photo-taking great.
DAY 8-9 Explore the wonders of Athens! Rick Steve website has great tours, also step-by-step FREE downloadable audio guide to Acropolis, Agora, AND the Famou Arch Museum. Study his website borrow guides from Library ... Prepare Prepare -- it will enhance your experience in Athens AND islands.
DAY 10 - off to Airport! Please do NOT book departure at 6 AM... u need to be in line 2 hrs ahaed, trip to airport is 1 hr; you'd have to leave hotel 3 AM - it would ruin your last night. Plenty of flights 10 & 11 get u to US by 4pm, home to chi by dark.