Mystic Seaport is a giant outdoor maritime museum. The wedding is actually taking place at the Seaport, but we recommend spending some extra time exploring, especially if you're into old boats. (You also get discounted admission during our wedding weekend!)
Williams Beach is a small park on the coast where you can take a dip. It's not too far from downtown Mystic, but you may need a car to get there...
Olde Mistick Village is an eclectic mix of tourist kitsch. You can browse through strange specialty shops (like Bestemors Scandinavian Imports and Sticky Situations, the artisan honey store), stop at the Bleu Squid for a cupcake and a grilled cheese sandwich, or tie one on in the Jealous Monk's beer garden.
Vault is a nice little independent cafe inside Olde Mistick Village--a quick walk from the Hilton and the HoJo.
This place is magical (if a bit fancy). We recommend lunch at the bar: oysters, chowder (they'll make it New England or Rhode Island style), bread and butter, and chardonnay. There's also the treehouse out back, which is an outdoor bar annex with a more casual clam shack vibe.
The Spicer Mansion is a super swanky inn and restaurant with a speakeasy in the basement. Old-school cocktails with a modern, exclusive twist.
Grass and Bone is more of a lunch spot with meat-forward fare, in the same restaurant family as Oyster Club and Engine Room (all fantastic).
Engine room is a boisterous upscale-pub type place in the same family with Oyster Club and Grass and Bone; the food is delicious, and the beer list is solid.
This is an old-school New England seafood standard with solid food and amazing views of the river.
Legit riverside clam shack: picnic tables, plastic forks, river views. There's an ice cream shack next door, too.
A standard basement Irish Pub--as unfussy as it gets.