We love to go to the beaches, hike, try new food and shop. Possibly do
a cooking class
You're better off avoiding Santorini if that's what you're looking for.
Beaches:
The few famous beaches on Santorini are made of black sand and dark pebbles, they're hotspots for Instagrammers rather than swimmers. Plus, they can be difficult to access. The famous Red Beach that everyone wants to visit is officially closed to the public due to the risk of cliff rockfalls. The beaches on the east coast are as generic as they come, the kind you find along any Mediterranean shoreline. No need to go all the way to Greece for that.
Hike:
There are a few hikes other than the famous Fira-to-Oia trail, which is actually just a 10-kilometer path running alongside villages and luxury hotels. In high season, it's packed. In the shoulder season, there are still plenty of people, including quite a few of what I call "flip-flop hikers." To get some peace and quiet, you have to head out very early in the morning (before 8:00 AM), which also helps you avoid the harsh sun.
Shop:
Santorini welcomes over 3 million visitors a year (for 15,000 permanent residents), so 90% of the stores you'll see sell trinkets for tourists and cruise ship passengers, while the others sell some local artisanal goods (which you can find for half the price on other islands).
Food:
In Santorini the food is identical to the other Greek islands, but much more expensive. That said if you're a wine lover Santorini wine is world-renowned. You can visit wineries and do wine tastings. There's also a tomato museum!
You didn't mention history, an important archaeological site on Santorini is the Minoan city of Akrotiri, which is worth a visit.
To sum it up, when it comes to beaches, hiking, and food, Paros, where you also seem to want to go, is a much better spot.
Naxos could also be an option, although I don't entirely agree with Janet regarding beach accessibility by bus, unless you're only comparing 3 or 4 beaches out of the around 25 to 30 available.
The big problem with Paros is that there are too many French people. :-)