If I had a dollar for everyone that tells me "You really should be careful riding your bike here, I would never do that" I'd be rich!
Riding a bike in New York can be dangerous but it's also really rewarding. It's empowering to ride a bike here and although I've been in lots of sketchy situations I will forever ride. I think its the best way to see the city.
Bike shops can be a total rip off in this city, so I have found some low cost solutions:
Times-Up.Org is a great resource for free organized rides as well as low-cost classes
on how to repair your bike. Every Tuesday it's Ladies Night and they host a women only repair course for free.
If you don't have a bike yet I recommend Recycle-A-Bicycle or Craigslist for buying your new wheels. You'll get a used bike for next to nothing and voila-you're ready to enjoy the City.
May is Bike Month so get out there and start pedaling and remember to wear a helmet!