The history
The Stonewall Inn on Christopher Street in Greenwich Village is the site of the June 1969 uprising widely recognised as the catalyst for the modern LGBTQ+ rights movement in the US. The inn still operates as a bar; the surrounding Christopher Street area and Sheridan Square are now a National Monument. It's worth visiting as a historical site regardless of whether you're LGBTQ+.
The neighbourhoods
Greenwich Village and the West Village — the original centre, and still home to the Stonewall Inn, a strong LGBTQ+ bar scene, and the most visible LGBTQ+ community presence in Manhattan. Christopher Street Pride each June is the emotional centre of the city's Pride celebrations.
Hell's Kitchen (Midtown West) — the current main bar corridor for gay nightlife in Manhattan, particularly along 9th Avenue in the 40s and 50s. More nightlife-focused than the Village, with several large bars and clubs running late on weekends.
Jackson Heights, Queens — one of the largest and most visible South Asian and Latin LGBTQ+ communities in the US, with its own Pride parade (one of the largest in Queens) and a vibrant, distinctly different scene from the Manhattan-centred narrative.
Brooklyn — Williamsburg, Park Slope (historically known for its lesbian community), and increasingly Bushwick and Crown Heights all have significant LGBTQ+ presences and bar scenes.
Pride
New York City Pride takes place in June — the anniversary of the Stonewall uprising. The main march runs down Fifth Avenue and is one of the largest Pride events in the world. NYC Pride also encompasses a week of events, parties, and cultural programming across the city; the Heritage of Pride organisation manages the official calendar. Book accommodation well in advance if you're visiting for Pride weekend.
General safety and climate
New York City has among the most comprehensive LGBTQ+ protections in the US, and public displays of affection are normal and unremarkable in most of the city. The NYPD has a dedicated Hate Crimes Task Force, and reporting mechanisms exist specifically for bias incidents. Like any large city, situational awareness is sensible, but the overall climate is welcoming.