In a week I'd stay on Sao Miguel alone. We did a vacation there this past summer with kids a bit younger than yours, and it's very easy to do everything on your own. We had 5 full days and that was not enough. A week would be better.
Definitely rent a car. We rented an apt on the edge of Ponta Delgada for our entire stay and really enjoyed it. Roads are good and it's easy to get around. The whole island feels enchanted. I relied heavily on this blog and his advices are spot on, although you'll need a lot more time than 4 days to do everything he suggested:
https://www.geekyexplorer.com/sao-miguel-island-in-4-days/
Our whole family loved Ponta da Ferraria. We easily spent an afternoon there, having brought a picnic. Praia Moinhos was another favorite spot. It's a beautiful black sand beach with a charming cafe, lifeguards, clean bath and showers, and even free sunscreen. Our kids could easily spend a whole day on that beach, if we did not drag them away to hike to a nearby waterfall, which was gorgeous. Gorreana Tea Farm was another highlight, and it's very close to Praia Moinhos. We got there near closing time but the trails through the farm were still open. Shepherds had released goats to graze in the tea farm and the entire hill was filled with the melodic sounds of goat bells. The sceneries were sublime nearly everywhere we went. If you go in late June/July, you'll see lots of hydrangeas. Always pack a picnic or two, and enjoy the many view points. There are many beaches, hiking trails, and waterfalls. Beaches that are rocky typically have wooden platforms for ppl to sunbath on.
Have fun planning!
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