Been going to ACK for over twenty years–here are a few suggestions. If you just want to hang out near the Boat Basin, it’s easy to walk to restaurants. Straight Wharf, Ventuno, Lola 41, and Proprietors are great for dinner. Company of the Cauldron is a fun place that’s also downtown. There are two seatings (not sure about off-season) and the food is served family style. You can see the menu before you signup. The Brotherhood, The Clam Shack, and B-ack Yard Barbecue are more casual restaurants in town, which are good. For breakfast, go to Black-eyed Susan’s.
Definitely check online to see what restaurants are open. Some of the places above, might not be open in April. Island Kitchen is one place that is mid-island and open most of the year.
If you need groceries there’s a Stop and Shop in town and one mid-island, but better choices IMO at Nantucket Fish and Meat or Bartlett’s Farm. The latter two are not near the Boat Basin.
If you want to explore beyond downtown, there are bike paths all over the island–rent a bike or bring bikes over on the ferry. You can walk on the bike paths too. I’d definitely go out to Sconset and do the Bluff Walk and Sankaty Head Light House (you can’t take bikes on the path–it’s too narrow). There are amazing views of the Atlantic. There are other great walking trails on island as well. You can get trails maps online (Nantucket Conservation Foundation). Tuppancy Links is a great walk.
IMO, the beaches are the best sights in Nantucket. My favorite place is Great Point, an amazing beach/conservation area. You would definitely need a bike or car to get there. Jetties Beach is close to downtown and it’s fun. There’s also a restaurant there (my kids love it). I don’t think the food is good, but the views are amazing (most of the seating is on the beach).
The off-season is one of the best times to go to the island–no crowds!!