Hi Stan -
Actually I moved from Texas to Long Island and currently work in Manhattan.
The best way to get from JFK to Times Square is to take the AirTrain (it's about $5/person) to the LIRR’s Jamaica Station (Long Island Rail Road). From Jamaica you can take a train straight into Penn Station – you have two options here as well, there is a subway train that will take you straight to Penn that has a lot of stops but is much cheaper, or you can take the LIRR straight to Penn which is more expensive, but much faster (I forget the subway train letter, but they should be able to tell you at Jamaica).
From Penn Station, the walk up to Times Square is *worth* the time, unless you have luggage of course (or it's raining, sleeting, snowing, etc...). Taking a cab from Penn up to Times Square should only cost you ~$20 or less. Penn Station is also directly underneath Madison Square Garden, so it might be worth popping your head out from under the ground to take your first touristy photo!
If you're "hotelling" up in the Times Square area, I would highly suggest getting to know the subway system because all of the hotspots you mentioned (except for Broadway) are either in Uptown or Downtown (Times Square is considered “Midtown”).
A 7-day unlimited ride MetroCard is only $24/person, which compared to taxi prices is a steal. It’s been a long time (3+yrs) but I seem to recall that a cab ride from Penn Station to the Financial District (Downtown/South Ferry/Statue of Liberty/Battery Park area) cost around $40 for just myself.
Link to MetroCard pricing. The subway system can be very daunting, so be prepared to hitch a cab ride if you get completely lost. I will say that NYC has a ton of NYPD walking around, and everyone I’ve talked to has been very helpful. If you find one that’s not, don’t worry, you’ll probably find two more one block down. I think they teach the police to recognize the deer-in-the-headlights look of lost tourists as part of their training.
If you do plan on taking the subway, get to know
this map very well. You can pick up a hard copy at any subway station (you might even find them at the airport) – unfolded the pamphlet is huge and makes a pretty neat souvenir poster in itself. I had a cousin who came in from Tokyo and she took handfuls home to give to her friends.
For Broadway shows, I highly recommend seeing the Disney shows if you can get tickets. While you probably already know the stories, both Beauty and the Beast and The Lion King are highly rated – I haven’t heard as much about Tarzan, but honestly everyone who I’ve talked to who has gone to a Disney show said that it did not disappoint. Mamma Mia and (for something a little odder) the Blue Man Group shows are also usually highly spoken of.
Last but not least, DRESS WARMLY! Winter is in full effect here and the average nighttime temperature has been around 20 degrees (daytime is in the low 30’s). The wind in Manhattan is killer, and it will flip umbrellas inside out without a second thought!
I hope that helps Stan, if you need any more information let me know and I’ll tell you what I can!
Time Square was the funnest place there. The saddest place on the other hand was seeing ground zero were the twin towers used to be.