The last time I was in New York City for a few nights, I bought a week-long metro pass for the subway. I figured if I was going to be bopping around the city from Chelsea to the FiDi, over to Greenpoint and back for the meetings I'd booked, my best bet for avoiding traffic would be to do as the locals do. Sure, it's intimidating sometimes - and fairly likely you'll miss a stop or go the wrong way on a train - but to me, that's all just part of the experience. I've gotten into the habit of trying to take public transit when visiting a new city - here's a few reasons why:
1. It's usually the cheapest way to get around a city. This may seem like a no-brainer, but come on. A week-long subway pass in New York cost me $32. I probably used the trains 6-10 times a day while I was there. If I had tried to use Uber or Lyft, I could have easily spent $32 on one ride from Manhattan to Brooklyn. If you're a traveler on a budget, public transit is almost always the way to go.
2. You get a glimpse into the everyday life of locals. Locals take the trains, and the buses and the trolleys. When you take public transit, you sit alongside the people who live and breathe the city you're exploring. You notice new languages, trends of footwear, ways to carry belongings, or whether people read, or listen to music, or just gaze out the window. And by observing the locals, you're observing the heart of the very city you're in.

3. It helps you understand the geography of an unfamiliar city. When you're constantly looking at transit maps and routes, you quickly determine which way's north, and which way's south - and which train or bus lines run east - west or do a loop around the center. And let's face it - it's much harder to get lost when you know (generally) where you're going.
4. It's environmentally friendly. Taking the train with hundreds of other people is definitely more eco-friendly than having that UberX to yourself. And heck, walking is even better!
5. It makes you a better traveler. I'm a firm believer that those who travel earn a certain kind of knowledge that can't be taught in a classroom or read in a book, or even shared from one person to the next. This kind of knowledge is first-person only, and it bolsters your common sense and problem-solving abilities. It beefs up your will to understand and appreciate a city, a culture, a history. And the interactions you have asking that friendly but distracted woman on the subway platform if you're getting on the train in the right direction will make you a better traveler. The frustration you feel when you realize the train you're on is click-clacking over the Williamsburg bridge, when you had no intentions of heading to Brooklyn at all, will make you a better traveler. The satisfaction and relief that you feel when you take your shoes off after a day of walking up and down the side streets of an unfamiliar city will make you a better traveler.
Take the train. Or the bus, or the ferry, or the tram or the trolley. You'll open your eyes to a part of the city you'd otherwise have never seen.