Sorry, back to the original question:
Like Theo said, Brotherhood is good - best calamari around. Best chowder, tough to say. A lot of people rave about Captain Tobey's - on the fast ferry on their way home to America. It's good - don't know I'd call it the best. I had a seafood chowder at Straight Wharf that blew me away, and if you're up for a not too fancy - and not too cheap, but not too bad - dinner, Brant Point Grill at the White Elephant has a lobster a la plancha that will blow you away.
I'd cab it up to Cisco brewery - always a good time. I would not rent a moped - for half the rental cost, and a fraction of the time, you can get the same result by paying someone to beat you repeatedly with a 2x4, work you over with a belt sander, and hit you with their truck a few times. Be leery of cobblestone poisoning in the late hours leaving the Rose and Crown - that hurts the next afternoon.
Speaking of cabs, there is one and only one cab you want to use, and you'll have to call ahead - and that's Chief's Cab. I just deleted a bunch of Nantucket numbers from my cell phone (since I don't live there anymore) or I'd give you the number. He's the retired fire chief, great guy, knows everything - and I do mean everything - about the island, and one of the friendliest people on Earth. He actually has a Nantucket Nectars cap about him. Taxi fares vary on Nantucket even though they are set by the town/county, and people occasionally gripe about overcharging when their fare is different than usual, but 99% of the time it's because most drivers don't normally charge all the 'add-ons' and full fares. Rate sheet will be posted either way. I've never heard of someone truly getting overcharged. Save yourself any hassle and do yourself a favor, though, and call Chief if you need a cab. He's a character you don't want to miss.
Be careful if you're riding a bike along the roads - great bike trails there, but when the tourerists are on-island, look out on the roads! I almost feel bad saying this, because I know a couple CT people that are great people and NOT like this, but if you're pedaling along and see anything with CT plates coming, dive as far from the roadway as you can.
Kitty Murtaugh's for the triple tenderloin.
The Bamboo for a good bar fight. (Seriously, I wouldn't go there)
The Hen House for breakfast - people will tell you Downeyflake, it's not all that. Tourists love that place, though. Left the better breakfast at Hen House wide open for the rest of us.
Lifesaving and Shipwreck Museum - this is one thing you DO NOT want to miss. Great museum. They HAD a fantastic exhibit with my ugly mug in a few pictures, but I guess it was making children cry so they rotated it out. It's up the Polpis Road. Worth the ride, it's not far. Nothing on Nantucket is far.
Man, I miss that place....