Definitely cab it!! If you're going with more than even 1 person, the difference in costs between a cab and the train/shuttle is really meaningless, and the drive into the city is much faster. Little tip: after you pick up your luggage, make sure you head out to the actual cab stand and get a yellow cab. DO NOT go with one of the gypsy cab drivers who stand in baggage claim looking to "help you" with a car. You WILL get ripped off, guaranteed!
As far as museums, it really depends on what you like. I typically tell visitors that my personal 3 "must do" museums would be the Museum of Natural History, MoMA, and the Met, but there are so many, you are sure to find one that suits your tastes. Have fun!