As a US Citizen, you can enter Thailand via air from anywhere and normally get 29 days on arrival (the current day is counted as one of the "30" days). Technically, you do need an air ticket out of Thailand (which you will have), although it is not usually enforced.
I enjoyed a short stay on Ko Phi Phi. I took the boat there from Krabi (not Phuket, as you would be doing). June is low season so you probably won't even need a hotel reservation ahead of time. The island boat tour originating on Ko Phi Phi is great. You see some world class scenery and visit 2 or 3 other islands with awesome beaches. On Ko Phi Phi itself, there is basically no Thai culture, it is a foreigner beach destination. Anyway, 3 nights there was enough for me (so 2 full days). One full day doing that island tour. And another day just putzing and spending time on the beach -- I seem to recall there was some kind of hike we did, also. But with your limited time, you may just want to stay in Phuket.