If there's any possibility of going on a weekday instead, I'd do that. Weekends are mobbed until the weather really cools down. We live very close, are members, and go at least once a week, usually a weekday (excluding Wednesdays). My son requested to go today (Sunday), and it was too crowded to park. Normally, we'd head to the Botanical Gardens a few blocks over, but they were also super-busy due to the Kiku Chrysanthemum exhibit (very beautiful and worth seeing). Our favorite time to go to the zoo is actually on a snowy weekday in the dead of winter--it's quiet, beautiful, and the animals (especially the tigers) are VERY active!
If you can't make it on a weekday, try to get there as soon as the zoo opens on whichever weekend day you pick. First thing Sunday morning would probably be best (I think even on Sundays, they open at 10 a.m.)
I wouldn't try to do everything in one trip--there are so many interesting exhibits, you could easily exhaust yourselves by rushing through to see all them. I'd advise to pick a few of the best (Congo Gorilla Forest, Madagascar, Tiger Mountain, and World of Darkness are our favorites) and just really take your time to enjoy them. Oh, and don't forget the children's zoo--our three-year old loves it.
Look on your zoo map to find out when the feedings and enrichment sessions take place. The tiger enrichments are usually three times a day, at 11:30, 1:30, and 3:30--really worth seeing, but get there early, as it gets packed with people. Also fun are the sea lion feeding at the renovated sea lion pool and the bee eater feeding in the World of Birds.
Oh--I almost forgot the Bug Carousel--really cute and fun! My son has become great pals with Josephine, who runs it on weekdays.
Feel free to PM me if you have any other specific questions, and please let us know how you liked it!