arseNal is right. Me, I like the Third Avenue Bridge, but it can be tricky to find. And worse, when you’re heading back out of Manhattan, you have to take another route if you’re in a commercial vehicle.
I recently rented a commercial van and drove across the Triboro Bridge. The toll is $4.00, I think. They asked me how much the truck weighed, and I had no idea. The guy at the toll let slip that they were making sure it didn’t weigh more than 7,000 pounds. The van was loaded with stuff, and they didn’t search me or anything, but if you’re going through a tollbooth, I’d recommend you make sure your car papers are in order.
Once you get into the City, Second and Third Avenues are very good for zipping right through, even if you’re in a large vehicle. But from the looks of your map, it seems you might want to get over to the West Side and head most of the way down Broadway, maybe hang a left at Times Square or 40th Street, then left again onto Sixth Avenue. The traffic there is hell, and you might have to go farther downtown, because some of the Midtown streets don’t let you turn onto them. I’m not sure if you’ll have this problem.
Good luck. When it comes to driving in Manhattan, my advice is: avoid it if at all possible, and make sure there’s beer wherever it is you’re finally going to stop driving for the day. But then, I’m from a small town in Pennsylvania, where they don’t drive like they do in this place; you might be better acclimated.