I just did an Athens/Santorini trip last month in the supposed off season. Athens was pretty cool, but no more than a three day visit unless you're a huge ancient history buff. I took a quick cheapish 45 minute flight on Sky Express, a Greek airline, to Santorini. I stayed in the beach town Kamari which is a 20 or 30 minute car/ATV ride to the main bleached white cliff towns Thera and Oia you see in all the pictures. Most people rent small cars, ATVs, and mopeds to get around the island.
I rented a buggy for a day and drove to Oia and then Thera and both were absolutely packed with tourists. I think I got to the towns shortly after one or two cruise ships disgorged their thousands of guests onto the island. I walked the length of both little towns in about 3 hours. It got boring really quickly and I longed for the beach town I was staying in. The beach town I stayed in, Kamari, which is one of two beach towns on the island, is made up of large black rock beaches. It's pretty to look at, but make sure to have water shoes because it hurts like hell to walk on them. The water is very nice, cool, and clear. Kamari has a ton of restaurants and bars and was actually pretty cool to stay in. The Greek food was mostly excellent.
But if you're coming to Santorini because you think the old white towns are going to keep you occupied for days with tourist activities, you'll be sorely disappointed. I think your kids would likely get bored pretty quick as well. So if your wife insists on going to Santorini, at least stay in Kamari or Perissa to keep your kids happy and occupied.