New York and LA are the two biggest tourist spots in America, so there are heaps of things to do... so probably best to do some "tourist" type search terms in google, and add in your interests (like movie studios, themeparks, sporting events, guided bus tours, monument, sightseeing, or even something specific like observation deck to find which buildings are open to the public for a fee for views of the city).

(check out viator.com as it often has tours and tickets to some of the real touristy things in many major US cities)

To see the various tours, themeparks, landmarks and other places of interest I've done in LA and NY since 2007, scroll to the bottom of this page and look at the different sections in the "Special Events" that relate to America. I don't do too much exploring beyond the very Touristy places, but there might be some things there that you might add to your list of things to do, or things you didn't find in a google search.

As UltraMarginal suggested, rent a car if you plan to be there for a while and need to get around, or need to get to something that isn't covered by public transport.
Taxis are pretty cheap over there compared to here, but if you are going for a while and want to do a lot of things, a rental car will be the best thing.... just be aware that they drive on the "wrong" side of the road, which is a bit of fun to get used to at first.