If it's a weekday, I'd probably go bus-subway (M-60 to N or W train to either 5th Ave or Times Square; or Q-33 to either E train to Times Square or F train to Rock Center). The cab will probably get you to Manhattan quicker, but once you're there, you may sit in traffic trying to get across town in the middle of the day. You could speed things up slightly by taking a cab to the subway instead of the bus. Others may have different opinions, though.

If it's a weekend, I'd take the cab the whole way. Subways run less frequently on weekends and often have maintenance-related reroutings and schedule changes that make it tough to plan ahead. Plus traffic is usually less of an issue mid-day on the weekend.

For lunch, I'd probably walk down 9th Avenue in the 40's and see what appeals. There are tons of little restaurants with every kind of ethnic food imaginable. For a decent slice of pizza, Ninth Ave pizza (I think that's what it's called) on 9th just below 53rd is a decent neighborhood joint (lacking in presentation, but big on flavor). For every kind of burger topping imaginable, Island Burgers and Shakes on 9th just above 50th (or 51st?) is good. For solid Sichuan fare, Grand Sichuan at 9th just above 50th (flip to the back of the menu, past the "American Chinese" options). For authentic Thai, Pam Real Thai on 49th just west of 9th (the other Thai places in the neighborhood--there are at least one or two on almost every block of 9th Ave--have funkier atmosphere, but don't compare food-wise). For the Puerto Rican diner experience (eat the counter if you're solo), El Deportivo on 9th at 48th. For great nouveau Mexican-fusion (a bit more expensive), Hell's Kitchen on 9th just below 48th. For health-conscious fast food (try the ostrich burgers), Better Burger on 9th just above 42nd is great. For dirt cheap hot dogs, there's a papaya drink and hot dog place on the southwest corner of 42nd and 9th (but I don't eat hot dogs, so I'm not sure if this is one of the "good" places). There are also about 100 more restaurants on 9th between 42nd and 54th or so. Oh, and if you want superb Israeli-style falafel (closed on Saturdays), check out Azuri Cafe on 51st just east of 10th Ave. My Israeli grandparents are obsessed with that place.

Ninth Ave is an easy walk to/from Times Square and Rock Center and the various trains that head out to Queens (the E train is closest, with stops at 50th and 42nd Streets along 8th Ave).