I don't think 113th street is that far north, it's near the north end of Central Park. I just looked up the 5-line schedule and it would take 20 minutes to get from 125th street to Bowling Green (which is the far south end of manhatten, like if you were doing to Ellis Island or the Statue of Liberty).
Based on that I would guess things like Times Square, the theater district, and rockefeller center would be about 10 minutes, and the museums are mostly on the upper west side so you'd be close to those. Like you said the trickiest part is that there's no east-west subway to cross to the westside for a while. The busses are good and frequent but you'd definitely have to split up to get 40 people through central park to the westside if you did the museums there.
As for safety, it's a little hard to say. The upper east side is where all the super rich people live, you're most at risk of Robert Downy Junior falling off tha bandwagon and making a scene or something. But that is also pretty far north, close to Harlem, so I would guess you're a few blocks outside that uber-wealthy area.
Really, in manhattan these days I don't think there's anywhere that is unsafe.
And what kind of fancy school do you work at that can afford to send 40 kids that far?! I think the only field trip I ever had in school was to a water treatment plant
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