For breakfast, ignore the tourist obsession with the Downyflake. It's good, but nothing special. You wait an hour for a table for nothing special. You want to walk across the street/Stop and Shop parking lot/street to the Hen House. It's down behind the pharmacy. Best breakfast on the island. Try the eggs benedict with corned beef hash.
Best calamari on island is the Brotherhood of Thieves. Straight Wharf has great seafood chowder. One fantastic restaurant (bit pricey) is Brant Point Grill, at the White Elephant (right by the boat ramp and Children's Beach - walk out of downtown towards Brant Point, hang a right on Easton St., it'll be on the right) - lobster a la plancha that will blow your mind.
Best pizza you will ever eat in your life? Pi Pizzeria. It's in the mid-island, also, close to Hen House.
Cisco Brewery - don't miss Cisco Brewery. I'd just take a cab there.
And speaking of cabs on island, they are cheap, it's rare for them to overcharge tourists (they've been known to habitually undercharge residents because they know they'll make it up on the tip, so when they charge full rate, people whine they got gipped). But still, if you're calling for a cab rather than grabbing one in the line at A&P parking lot or Steamship, there is one and ONLY ONE cab to call. Chief's Cab, 508-284-8497. Trust me on this one - he's the best cab driver around, guaranteed to make you laugh, and the only cab driver you''ll ever ride featured in his own Nantucket Nectar's cap (seriously).