So far no one has mentioned the lobster pound at the dock in Guilford, around behind the parking lot for the Guilford Mooring. Avoid the Mooring, but go to the lobster pound (weekends only, I think). Local lobsters, lobster rolls, and a few other fresh shoreline specials.
505 Old Whitfield St, Guilford, CT 06437
There is a limited amount of local seafood available in Branford. There is a lobster pound in the old foundry complex by the river off of Indian Neck Ave. You can pick up lobster and clams there. Across the river is the Lobster Shack which has a nice lobster roll. You can check out Bud's for a larger selection of seafood that's not necessarily local. You can try and catch your own. Snapper blues are there now. If you are staying by the water, try clamming too.
There is not much in the way of farmers markets, but there are a lot of farm stands. We go to the one on 146 across from Lenny's. You should keep Lenny's in mind for steamed lobster, fried clams and chowder. If you go to Guilford, Bishops has a great selection of local produce and fruit, much of it from their own farm/fields/orchards.
7 Indian Neck Ave, Branford, CT
I buy fish from two sources that are around Branford - Dolan Brother Shellfish and Gambardella's Fish - but I buy from them in New Haven and Milford. The Dolan Brothers sell lobster, clams and oysters at the Wooster Square Farmers Market in New Haven on Saturday mornings - great stuff, very fresh. Lobsters go for about $!0 ea; this may be (or may not be - I haven't checked lately) higher than supermarket prices, but these critters haven't been hanging around a fetid tank for two weeks. Oysters are $1 (I think), clams go for $6-8 a dozen. All are very fresh, very good.
Gambardella's sells fish on Saturday mornings and Wednesday afternoons at Robert Treat Farm in Milford; but they have a dock in East Haven (neighboring town to Branford) and seem to be very accommodating, so maybe you can work something out. I've bought flounder, amazing scallops, wild salmon, local (Stonington red) shrimp and more for a couple of years, and have just had great fish from these people. Sometimes the shrimp are from Carolina, but they're flown up and still excellent. Gambardella East Haven phone - 203-467-2800
I'm not local to Branford, and others may have other and/or better options; but enjoy your stay, and I hope you enjoy our fish!