There is so much to see I'd recommend bringing a guidebook or researching the exhibits first! We did the Rick Steve's walking tour, which took us through the major sites of the Egyptian, Assyrian and Greek exhibits. Having the guidebook was great because Rick provides history on the sights that you are seeing and directs you to the most interesting and most famous attractions.
Make sure you see the Greek exhibit while you are there! The Elgin marbles were beautiful and fascinating to see. And we found this exhibit a bit less crowded than the other exhibits we saw.
A donation is suggested, but there is no one asking for them, just donation boxes placed around the museum, so you don't feel pressured to donate.
Make sure you give yourself at least 3 hours to see the museum even if you are just hitting the big attractions. There is a lot to see and the museum gets very busy! You could take a week and still not see all the whole museum.More