Like most things in life, it depends on budget. There is nothing like staying in NYC and being there at night. It would make the trip so more memorable. Locally there is nothing. After all, it's New Jersey, not NYC

It's easy to spend other peoples money so I say stay in midtown. You'll have trouble near the holidays getting a room let alone at a bargain but you really want a September or October game to avoid a cold weather game, whoch could be unpleasant at times and may get a glare or two from the wife.
Watch the hotel prices, they vacillate wildly throughout the year. The best bet for deals is Hot Wire, but you have to actually buy the room before knowing the hotel you stay at. You do get the price, star rating, area and user recommendation (sort by that first). We've gotten some amazing deals (the Waldorf at 75% off) and some disasters (a hotel room so small that Tyrion Lannister would not have fit in it). Once you decide, repost and people will give you all sorts of suggestions
To go to the game, you would probably take the NJ transit under MSG. we tried it once and disliked it but that's another story. I think that there are buses somewhere also but not sure if they still run. Some budget minded folk stay in Jersey City and use the PATH train. Not sure how that works either.
If you get to the stadium early enough perhaps you can hook up with someone's tailgate, which is often the best part of the day.
For tickets, it will depend on the opponent, obviously. There are cheap seats ($50 each) that you can get from the Jets but they are in the Upper end zone and waaaaay too high. Fan Snap will give the your best bet at resales (includes Stub Hub and all the other majors in one search)
New Giants Stadium is not the most interesting stadium to visit, but once in the gate, you'll still be outside and there are a few things to look at and activities, such as kicking an extra point, throwing footballs, food trucks (the better of the mediocre foods in the stadium) before you go up.
Once you decide to go to your seat, be careful that you use the right access as sometimes the clubs block off access to the rest of the deck (I.e. Mezzanine)
The earlier you plan, the better chances you'll have for researching the best prices and tickets. Schedule in 2-3 weeks. Give yourself a couple of choices if you can.
Hope this helps a bit