To get to the Museum of Natural History you can simply take the cross town bus that runs on 34th st.
You get an automatic free transfer to the subway. You get off at 8th Ave, and take the c train to 81st or the A to 59th and switch to the B or the C at that point.
The subway station itself is a worthy destination--it has marvelous mosaics of dinosaurs, paw prints, skeletons, prehistoric birds, etc. Its very artistic and colorful. Your kids might like to ride in the first car so they can see the train going into the tunnel. If you are in a rush, you can always cab it to the museum
and take the subway back. Uptown in the morning isn't too terrible and downtown in the pm is ok.
There are lots of decent restaurants on nearby Columbus Ave; weather permitting, you can also walk through the park in the early pm and take a downtown bus on one of the downtown avenues. Be sure to check out Central Park, its beautiful even in the winter.
If you love the subway tiles, be sure to visit the 59th and Lex station and the 42nd and Broadway ones.

ttt for aduchamp subway do's and don'ts...