You have two options.
If you fly into Logan and stay in the city, then you will have plenty of places walking distance. Back Bay is awesome, just a lot of places for you to go in the city. The one thing I would be aware of. If you are renting a car, not sure how that will pan out. Expensive to keep in the city, and during the week depending on the time of day, leaving and getting back into Boston can be an absolutely NIGHTMARE.
If you fly into Providence you can stay at fairly nice hotels. There are a couple Marriots/Westin in the city. The drive to Gillette is 25-30 min away. I'm sure flying into and out of TF Green is exponentially easier than Logan as well, if that matters. If you are looking for something to do during the day, rent a car. There isn't nearly as much in Providence as there is in Boston, however we have a beautiful shoreline, and if you are a foodie person there are plenty of great places to visit all over the state. Heading to Newport for a day (or two) would be my first suggestion.
As far as the game goes. Definitely tailgate. Its fun. A lot of locals will park at the lots further away from the stadium. For a first time goer, I would suggest getting there as early as possible and park next to the stadium. Its more of a nightmare getting out, but the location/being right there could be worth it. If the game starts at 8:30 i would suggest trying to get there for 4ish. Lots don't open up that early but its best to get in line. Chances are you'll hit traffic no matter what so why not go early to enjoy the stadium as much as possible.
As far as seats, i would suggest avoiding upper corner 300's. Lower 300 rows (especially the first 8) are actually pretty nice. Everything else is a great seat. I personally don't like endzone, I'd rather see the game left to right, but that is all speculative. I was in the endzone against the Ravens divisional game 2 years ago and it was better than i was expecting.
I wouldn't suggest booking your return flight for Friday AM. You won't get out of the stadium until 12, and probably another hour or more waiting in the parking lot.
As far as places to go, Patriot Place has a ton of placed. Most of them are blah... I go there to tailgate, bring my own stuff, grill, etc. However since you are traveling you can hit up CBS Scene, its their "sports bar" that is always packed during game. Again, get there early. Also, check out the patriots place website so you can get a lay of the land and a list of all the places/shops that are there. Davio's is overrated don't waste your money on a steak there. Bar Louie has good bar food. CBS has crap food but whatever drink more than what you eat lol.
Let me know if you have any more questions