Salem is worth visiting, it is very tacky in places with a lot of occult nonsense (ironic that the Salem witches which triggered all the woo were in fact not involved with the supernatural). There is a good walking tour based on the true story, an excellent museum and art gallery which are worth seeing.
How long will you spend in Boston? I usually recommend either the Duck tour or one of the trolley tours to get the lie of the land. The hop on hop off tours are good if you have limited time and want to see a few things. The Constitution (the oldest commissioned warship in the world) is interesting and free but you need to have your passport with you to get on to it. The Aquarium is excellent but very busy in Summer, Science Museum similar. The view from the top of the Prudential Tower is great and there is an audio tour that explains the main points of interest. The Museum of Fine Arts is a fantastic gallery but enormous, The Isabella Stewart Gardiner Museum is smaller, quirkier and cheaper (free if your name is Isabella, $2 off if you are wearing Boston Red Sox merchandise), there is an excellent audio tour there that you can upload to your phone.
You can take a boat trip round the harbour islands which would be good on a calm day, or a whale watching trip, though if you are going to Maine it would probably be better to go from there.
There is a vast array of food outlets in Quincy Market but try the Public Market next door for an amazing range of local food, there is also a cafe area which is less frenzied than Quincy Market.
The only place I have been in Maine is Boothbay, which was lovely (amazing icecream shop), the botanical gardens there are beautiful. We are planning a road trip up there this summer so I will be following this thread with interest.
I'm not sure about visiting LL Bean, it's just a shop, but if you feel the need for duck boots go for it! Freeport has masses of other outlet stores so if you want to shop that's the place to go, if you're coming from the UK be aware that the exchange rate is pretty dire at the moment so everything seems expensive in the US.