Baseball: Yankees and Mets
The Yankees (Bronx, Yankee Stadium) and the Mets (Queens, Citi Field) divide the city along lines that have little to do with geography and everything to do with identity. Yankee Stadium is easier to reach on the 4/B/D subway trains and is the historically significant venue; Citi Field is newer, more comfortable, and cheaper on average for a comparable seat. Both are genuinely good live experiences — baseball at a major-league stadium is one of the most distinctly American things you can do in the city, and regular-season tickets are far more affordable than most visitors expect.
Basketball: Knicks and Nets
The Knicks (Manhattan, Madison Square Garden) are the more storied franchise and play in one of the most famous arenas in sports. MSG is also a music venue, fight venue, and cultural institution in its own right. The Nets (Brooklyn, Barclays Center) moved to Brooklyn in 2012 in a move that divided and delighted fans in roughly equal measure. NBA tickets are significantly more expensive than baseball — expect to pay a premium for a good seat at either arena.
Football: Giants and Jets
Confusingly, both the New York Giants and the New York Jets play in New Jersey — MetLife Stadium in East Rutherford, accessible by NJ Transit from Penn Station. The NFL season runs September through January; regular-season tickets are available but playoff games sell out rapidly.
Hockey: Rangers, Islanders, Devils
Three NHL teams, three different arenas. The Rangers play at Madison Square Garden; the Islanders at UBS Arena at Belmont Park (Queens); the Devils at Prudential Center in Newark. Hockey is the sport that most locals who don't follow sports will still express a strong opinion about.
Getting tickets
Buy directly from each team's official ticketing site or the official venue box office. The same scalper-adjacent third-party dynamics that apply to Broadway tickets apply here — check face value first. Baseball regular-season tickets are often cheapest when purchased directly from the team's own site, sometimes even on the day of the game.