1. Staying Boston isn't huge and getting around downtown is reasonable with the T (= tram/underground) so, if money's an issue, I wouldn't spring the extra for a city centre hotel. I thought the Lechmere suggestion was good -- it's a couple of stops out of the centre on the Green Line. Easy access to Cambridge. (Cambridge is, in effect, a part of the Boston urban area -- sort of like calling Camden a different city from Westminster.) The places I know to stay in Cambridge are so-so and pricey. I'd trust the internet.
2. Travelling Top tip. Use the "Silver Line" to get in to town. You need to buy a ticket for the T (see above). There's a bus stop outside the International Arrivals building (Terminal E) way off to the right, as I recall. That takes you to South Station on a dedicated busway. Then you can get onto the subway and continue your journey. All on one ticket! As quick as a cab and much, much cheaper.
As for getting to the game, if you don't have a car, the train from South Station is the option. In its favour, the train is convenient and the atmosphere is fun. But it's not perfect. Takes an absolute age to get about ten miles and sometimes doesn't even manage to get there for kick-off! If I were you, I'd explore other options, so you can check out Patriot Place and the tailgating scene, but, if you end up on the train, so be it.
3. A Warning You can't take backpacks into the stadium and you can't leave them on the train. So make sure that you have everything you need in your clothing (and make sure you have lots of clothing -- the stadium is open and, if it rains, you can't have too many oilskins/plastic ponchos, etc. -- funnily enough, snow isn't so bad).
4. Places to go, things to do For me, the Museum of Science is 10/10. There are lots of great bars/restaurants. I know ones in Cambridge/Somerville (just behind Lechmere) but downtown there are a lot in the South End, which is pretty much the hip place to live these days (or so my friends who think they're hip tell me!) And there is great music of all kinds.
You'll have a marvellous time. People are courteous and friendly and the stadium atmosphere is wonderful -- very different from a European football stadium but a great experience nevertheless.
P.S. Remember to have your passport with you. You'll get "carded" (asked for ID) in bars very regularly -- indeed, I got carded at Gillette trying to buy a seven-dollar Bud -- and sometimes they won't accept a U.K driving licence. (I know, isn't it?)