For bike safety programs, talk to Wiley at Transportation Alternatives. Lukasz linked to it in his post.

For general getting around, you can get a free bike map at any bike store so ask your friend to pick a few up and send one to you so you can get familiar with the place. Lots of bike stores will rent you a hybrid type bike so do a search near to where you're staying and start calling or emailing to get a rate. Or you can pick up a used bike off CL or at Recycle-a-Bike.

As for other stuff to do, there's an all-female race on 6/20 as part of the BFF, but it looks like you'll be gone by then. On weekends, you can swing by Sarah Roosevelt Park and check out the bike polo matches. It's close to Chinatown so you can pick up some food while you spectate. And yes, you can do a loop of Manhattan, mostly on bike paths with some street. If the "Fun Random Night Rides" thread picks up again, you might be able to join us for one of those loops with a stop at Dinosaur bbq or Chelsea Brewery.