DC native now living in NYC, I used to take trips like this all the time with my mom when I was a teen, and still travel back and forth frequently. I will second what most have said to take either the Amtrak from Union to Penn Station, or any of the buses (most also arrive fairly close to Penn Station). Staying in midtown will give you great access to shows, food, and shopping (and will only be about 10-15 blocks from Penn, pretty easily walkable). For shows I recommend looking in to Peter and the Starcatcher, Wicked, or Newsies. You can check broadwaybox.com for discounts, or go to the TKTS booth in Times Square on the day you want to see a show to see what discount seats they may have available.
I also really like the Museum of Modern Art, I feel it has a nice blend of off beat Modern stuff and more classically modern art (VanGogh, Monet, etc).
Also, Rockafeller center has ice skating all the way through April, you get the great "New York" experience of skating there but it's not nearly as crowded as right around Christmas. You can also go to The Top of the Rock, which can be even better than the Empire State Building because you get the Empire State Building in your view
Also, for an idea of what may or may not be walkable, as a rule of thumb I generally find that I'm good to walk around 25 blocks in a stretch if need be, whereas when my Mom comes to visit, she's more comfortable with about 15 as her max in one stretch. She's healthy and plenty active, just not as used to city living. Oh, and don't forget that avenues are about twice as long as street blocks!