We did two weeks back in October. Stayed in Kapa'a which I thought was a good central location to do things on every part of the island.
Our favorite hikes were definitely Hanakapi'ai Falls (Na Pali Coast, 8 miles out and back) and the Alakai Swamp trail which I'd say is probably the most unique hike we've ever done (about 7.5 miles out and back). The swamp trail was super muddy when we went, so if you can take poles I highly recommend it. The Heritage trail was also pretty cool, although I wouldn't call it "must do."
Shave ice was a tough call because everyone has their favorite, but we found Hee Fat to be a can't lose. Once we found it I think we went back 4 days in a row. We also really enjoyed the
Porky's food truck in Waimea, and
Da Crack in Poipu.
If you're a dog person and looking for something a bit different, the
Kauai Humane Society will allow you to take one of their dogs on a field trip. We did this and really enjoyed giving the dog a day out, even though we found out that it hated both the water and cliffs (which Kauai has a lot of). We took it to the Heritage trail and after about 15 minutes realized the hike wasn't going to work out. But the dog rested in the back seat of the car for the 45ish minute drive back.
I don't have good advice for beaches as, unfortunately, our favorite beach ended up being the one on PMRF which most people won't have access to. Although I suspect that the fact that we had the entire beach to ourselves might be part of why we enjoyed it so much.
I honestly don't think you can really go wrong with Kauai as long as you enjoy the outdoors. Just the other day I was telling my wife I might consider doing OMY just so we have enough money to live there for a year or two after FIRE.