Although locals and
@snowleopard can help you much better with this issue, I allow myself to give you some suggestions (as well):
-First: You need AT LEAST an entire day for the visit. The zoo is big, for some maybe huge and has a great variety of species. So - if possible - plan for 2 days.
-If it is already crowded at the gates on opening time, visit the zoo clockwise, beginning with Desert Dome
-If it isn't crowded and if there is nice weather/sunshine, go counter clockwise thru the "Africa section" to the lions, take the skyfari back and do the "Africa section" again.
-Unless you are a train-fan, skip the "Omaha Train". You wan't see that much (animals).
-If you are hungry, I recommend the Thrulsen African Lodge adjacent to the elephant exhibit. Good food (on zoo level!

) and nice view.
-Skip the Childrens Adventure Trail if you haven't enough time left.
Ref. nearby zoological institutions:
It depends how far you would like to travel from Omaha.
Lincoln Children's Zoo and Lee G. Simmons Conservation Park are up to an hour away (just passed them by driving back but did not visit them). The next big(ger) zoo will be in Kansas City, which I guess is at least 3 hours away.