When looking at a map of the stadium, there is a section of red seats at midfield, these are the best in the house, with a box-like club that only people with these tickets can access. Unfortunately, these are generally reserved only for corporate partners, so unless you're buying from a sponsor of the stadium, these tickets will be difficult to find. Try the section on the opposite side of the field - you will have to be on the visiting team's sideline, but will get the best possible view. The nosebleeds on the visiting sideline are as bad as it gets at Gillette. During day games, the sun will be shining directly in your face for the entire game; but, that's picking nits, you're at an NFL game, every seat is an awesome one.
The New England Patriots games have been sold out since December 26, 1993, long before Gillette Stadium opened it's doors. When you get a ticket to a Patriots game, ANY seat is a good one. In addition, there are no bad views in the house, because every seat is focused exactly toward midfield. This detail offers the best possible sight lines.
Just to see a game at Gillette stadium is pretty special and since their aren't many open seats for games I suggest not being picky. Try your best to become friends with someone who can get you into a box suite which makes for a great experience.