Absolutely agree - use the train! You can also get trains that terminate at Hoboken and then take a ferry across to Manhattan, which is a fun way to arrive (apologies to regular readers, who've been bored by this tale previously)
For food, if you haven't eaten at a NY steakhouse, you haven't lived! My favo(u)rites are
Gallagher's (a former prohibition speakeasy, not far from Times Square - there's a franchise in Newark airport, too, but I haven't tried it) and
Keens (two blocks from Penn station and the Empire State Building).
Hope you aren't a vegetarian! Have you heard the one about the veggie who went into a restaurant in Wyoming?
Veggie: What do you recommend for a vegetarian?
Waitress: Son, I recommend you get the hell out of Wyoming!