The subways and buses are great, if you're not up for taking a cab! MetroCards are sold at all of the subway stations and work for trains and busses. You can add a specific dollar amount to pay for individual rides ($2.75ea), or buy a 3 or 7-day unlimited pass. Google maps works great for directions on public transportation too. If you get off track, don't be afraid to ask for help. New Yorkers get that not everyone knows their way around right away, and are aggressively helpful.
Everyone has their own opinion here - there are certainly plenty to try. We are partial to Thompkins Square Bagels, but H&H and Tal Bagels will also do the job. Savory spreads are the way to go.
If you absolutely must see Times Square, we understand. Our advice - hit the big sights that are close together (Midtown, or Lower Manhattan), and then find a great neighborhood to wander around. We're glad to share some recommendations. Plan your route to maximize time above ground (as opposed to in the subway); build in time to sit, relax, and eat; and don't try to do too much. You can't do all of New York in one weekend, so don't try. And you always have a host now for when you come back.
If you like classical art and paintings, The Metropolitan Museum of Art is "pay what you like". This could mean as little as $5, though you will see recommended amounts. If you like modern/contemporary art, The MOMA, Whitney, and the Guggenheim are all great, and charge an entrance fee of $18-$25. There are a ton of smaller museums around the Upper East Side, and they also charge entrance fees. But, if you're up for doing some extra digging - almost all of the independent galleries in Chelsea are free to the public when they're open.
It's everywhere. We recommend knowing what you're looking for. There are neighborhood shops where you can get a $4.00 slice to eat on the run (Scars on the LES), and 100-year-old sit-down brick oven pizzerias (Grimaldi's, Patsi's, and Roberta's are the classics). Artichoke pizza is amazing, but it's its own thing, and lately, there has been a wave of Detroit-style pizza places popping up around the city. They're good, but we're not in Detroit. You also won't find good pizza in any of the heavy-tourist areas (Midtown and the Financial District). Stick to where people live for this one.