The third-oldest major league baseball stadium in the country is still the best place to see a game. Granted the whole family wears the Pinstripes, so yeah, there may be some bias.
If you're staying in Manhattan (as most tourists do), the stadium is very easy to get to. Three subway lines go to the park, as well as a bunch of bus lines. If you're taking the subway from the West Side though, check which of the two trains is stopping at the stadium as they alternate depending on time of the day/week. Driving can be a pain. If you can navigate the local streets, do that and avoid the Deegan and Cross Bronx. Parking is also tight.
Prices are kinda steep and lines can be long, but there are few if any bad seats in the house. Bleachers are cheap, but can be rowdy, even though they don't sell beer in the bleachers. The bleachers are not recommended for fans of visiting teams. If possible, arrive a couple of hours early and enter through the left field gates. This is where Monument Park is, and the Yanks celebrate their stars of the past.
Some consider the stadium hot dog a must, but the stadium has a number of alternate meals as well. Along both foul lines on the field level are two food courts that offer chicken, roast beef, cheese fries, and hamburgers. Pizza and tacos can also be found in the stadium. The chicken fingers have become a family favorite.
There's not too much to do in the area besides the stadium. River Avenue under the subway offers a number of bars and baseball stores (also a bowling alley), and Stan's Bar is almost as much an institution as the stadium itself.