I am just going by your expressed interests (sightseeing, r & r, good food, culture/history AND hiking), and based on this, I'd second Charlene's & Julie's advice -- NAXOS. Not only is it an easier connect than Hydra, it has more of what you say you want. (Hydra for some reason, is almost the only isle beside Santorini that Rick promotes). Not much culture/history to check out in Hydra, the main deal seems to be, no-car ruling, so only donkeys -- but in other less-touristic isles once u go inland a bit, back roads are rustic, little traffic, even sheep flocks. If you are interested in swims, better look at alternatives.
I'd suggest fly to Santorini for 2 nights/3 days, then at 3:30 pm, take BLUE STAR ferry to Naxos. It has best beaches in Cyclades (about 15 miles golden sand, in a string, either stay right on a beach or take a frequent bus for 15 mins or so). Also best hiking in Cyclades; see this expert http://www.cycladen.be/NaxosEng.htm The port town is fascinating, a maze of lanes, history galore, forts, castles, ancient ruins, and great shops (at 1/2 price of Santorini). NAXOS is largest, most fertile isle, go inland a few miles and farms, sheep, vineyards ... In the little groceries near tourist lodgings, you often see a basket of fresh-laid eggs on counter. Despite great amenities, not overwhelmed by tourism because port doesn't accommodate cruise ships, airport doesn't take big planes. We vets usually recommend doing your Athens 2 days last, unless the Rick thing starts with them. You can return to mainland via BlueStar ferry (2 sailings per day, the 6pm one gives u a final Naxos day), or flying (small jet 1x per day). Of course, also possible to reverse this --first Naxos, then ferry (2+ hrs) to Santorini, then flight back to ATH airport.
It would be easiest to recommend which, if we knew exactly what the Rick tour entailed (i.e. start in Athens hotel? End where?).