Yes, meet people at the gym. Be friendly, and nice. Make friends. Sooner or later, you will meet someone who is heading out of the city to climb. Go with them. If someone is mentoring you, pay for gas, pay for food, pay for beers. Take time to learn to climb safely. Many new climbers clip a few bolts and then decide that they now know everything about climbing. There is so much to learn with regards to rock climbing and mountain sports, so never stop learning. Be humble. Don't rush out and buy the most expensive gear that you can find. Take some time to learn what you actually need. I have a bin of gear with alot of random pieces of hardware in the bottom that never get used because I realized that there was something better or more useful.

Sorry for the text wall.

Get a crash pad and get your ass beat on some boulders in Central Park. If it's nice out, you can meet other climbers hanging around the rocks. People there will be much more likely to climb outside, since they are already climbing outside. I found the gym scene in NYC were mostly gym climbers only, but that's completely anecdotal. I didn't seriously start climbing until I moved out of the city. 

This doesn't exactly answer your questions, but I recommend that you hire a climbing guide for a day. You'll have a fun day, get your ass kicked, and learn something. It might not help you find regular partners, but you'll at least have *some* experience outdoors when you do. There are some really good and fun guides in the Gunks, you just have to pay your money and get yourself (and your harness, shoes, helmet, chalk bag and belay device) up there.

Your friend introduced you to climbing? Wouldn’t he/she be a reasonable starting point for these questions? Unless they are also new like you?

Central Park bouldering is a good way to dip your toes into “outdoor” climbing.

I would also ask at the gym. Most gyms these days have sone kind of “intro to outdoor” classes, things like learning how to rappel, build anchors, etc. learning to lead in the gym won’t be much use for you for climbing at the Gunks, you really need to learn to place gear. But it is still something you should probably do.

There are also sometimes organized group outings...

There is a substantial investment in gear and a lot of things to learn before you can go on your own. Your best bet, while you are learning things that you can learn st the gym, would be to become friends with more experienced people and ask to go with them. 

I ended up joining the AMC and met a bunch of people that way. If you want to come to a top rope thing sometime send me a note.
There are two facebook groups for the Gunks, join those.

As Chase mentioned above, go climb at the Central Park Boulders. I recommend Rat rock, and, it you're up for it, Worthless Boulder. People are much more likely to take you climbing outside if they met you while climbing outside.  Plus you'll be climbing outside. Rat rock and Worthless boulder offer a pretty good time in and of themselves, especially for a very new climber. Just be prepared, as others have mentioned, to be humbled.

For the gym, work a project with someone. That always fosters intimacy. Oh, and learn to belay--both top rope and lead. That's probably the biggest barrier to getting someone to take you outside.

Good luck! I too found out I was a climber in New York City. Keep making an effort to get out of the city; its always worth it.

https://www.meetup.com/Rock-n-Ice/?_cookie-check=skXbD6gKSR1Ho-g-

This meetup group is how I met my crew many years ago, im not sure how or if it is still active but they occasionally would set up trad and anchor building classes, etc which is how I learned. There is a pretty huge financial entry to climb outdoors, you'll want to at least have a rope to offer to  share with potential partners. Plus personal safety gear, hemet, belay device, etc. Get a handful of friends together and hire a guide, it may seem like a big expense but its a great opportunity to have someone literally show you the ropes, provide all the gear and give you a fun and safe intro. Mountain Skills and Alpine Endeavors are both great.