I work in New Haven and have eaten at Louis Lunch many times. You can only get onions, tomato and cheese on your burger which would make you think that it might be a bit dry, but it actually is very juicy. The onions are cooked sitting right on the meat patty and they are cooked in some really old things. But it all works.
It is different then any other cheeseburger that you are going to get any where else and worth a try.
Pepe's and Sallys are both good and have a very long standing rivalry going on. Depending on when you are going, you may not hit a line. I drive by them everyday and fight the urge to stop and grab a pie for dinner.

There is a great pastry shop in the building next door if you are looking for some good Italian pastries for dessert.
Lenny and Joe's is soooo good for seafood. The Madison location is pretty much just fried seafood, but in the summer time they have an ice cream shop and a merry go round outside that the kids love. You go to the counter, order your food and wait for your number to be called. At the Clinton location you are seated at a table and have an actual waitress. The menu is much more extensive including broiled seafood, salads, and other non-fish items. Plus a full bar. My favorites there are the scallops and shrimp, just about anyway they cook them.
Bills Seafood is down the road in Westbrook, right next to the singing bridge. You can sit outside in season and have a table by the river. They are best known for their clams. Both whole belly and the strips. The only place I will get them, including Lenny and Joe's.
There are outlet shopping centers in Clinton and Westbrook. They both have a good selection of stores.