Just got back from a trip there. We spent a week on São Miguel and based our itinerary from this site https://www.geekyexplorer.com/sao-miguel-island-in-4-days/. We didnt' do any super long hikes, as most of the attractions (waterfalls, viewpoints, hot springs) are located pretty close to parking lots.
I'd recommend Pico da Barosa. At the top, you can see both the north and south sides of the island. We walked up to the top and then did some ridgewalking on the informal trails up there. This is a short hike, but luckily it's right next to Lagoa do Fogo. It is a short (but steep!) walk down to Lago do Fogo. Once you're there though, you can walk around the lake.
Our favorite hike was Sanguino-Salto do Prego. This one is slightly longer, with more of a jungle feel. Great waterfall at the end, this one is a bit of a drive if you're staying in Ponte Delgada though.
In terms of general hiking advice, you'll have to contend with the "notoriously variable" weather, which can mean rain and view obstructing clouds, especially in the interior areas. You will want to bring rain gear you can rely on/spend some time in. Luckily, there's webcams for all the major viewpoints so you can check to see if they are worth a visit or if you should hit them later. Even when the weather's good the trails can be a little muddy, learn from my mistakes and don't bring brand new white running shoes.