It's definitely an all day visit. The only absolutely must see (in my opinion) is River's Edge. It's probably the best exhibit complex at the zoo, with both older and newer exhibits. If you want to see the Children's Zoo without paying, arrive in the first hour the zoo is open. It has a few rare species like echidna, tree kangaroo, and Tasmanian devil.
I'll give a brief summary of all exhibits, dividing up them into the six areas that the map does:
River's Edge: animals from Africa, South America, Missouri, and Asia. Begins with South America (Bush dog, macaw, anteater, capybara, spectacled bear) with some great exhibits. The newest is for spectacled bear, and it does a pretty nice job. Africa is next, with black rhino, sacred ibis, African wild dog, cheetah, spectacles hyena, bat-eared fox, red river hog, bee-eaters, Nile hippos, and perhaps a few more. The newest exhibit here is for African wild dogs, all-around pretty sold again. Next is Asia, with Asian elephants and a great sun bear exhibit. Beloved elephant "Raja" was born at the zoo 20-some years ago, and he was the reason this complex was built. He is magnificent to look at. The sun bears have a new exhibit that is the second-best I've seen for the species. Missouri is last, with an outdoor pond and a large fish tank.
The Wild: a little bit of everything. Penguin and Puffin Coast is the most popular attraction here for normal visitors. I'm not too thrilled by it, but it's not bad by any means. The penguins have an open-topped exhibit, though it's tall enough to prevent most visitors from coming in contact with the birds. They have a quite successful breeding program for king penguins. Puffins have a similar exhibit. If you are very sensitive to cold temperatures, you won't want to stay for long. Outdoors is a Humboldt penguin and brown pelican exhibit. Next is Polar Bear Point. Odd viewing, smallish pool, but better than most polar bear exhibits. The train station is another attraction here- takes you around most of the zoo. There are two other stops but if I want to ride it all the way around, this is normally where I'll get on. There is an elephant habitat that can only be seen by riding the train. A big waterfowl lake is home to various waterfowl and flamingos. There are misc. exhibits for prairie dog and red panda that are decent. Lastly is the great ape complex, Fragile Forest. Every species of great ape except bonobos, and in quite good exhibits. If I recall correctly, in the indoor building for gorillas, there are Patas monkeys, though I may be thinking of a different zoo. The exhibits for all the animals in this section are average or better.
Red Rocks:
If you like hoofstock, this is the place for you. Nearly all of the zoo's hooofstock is here, representing some rare and unusual species. The exhibits are of varying quality, the main issue being space. Since most exhibits otherwise are the same, here's a species list: Bactrian camel, lowland nyala, ostrich, giraffe, Javan banteng, visayan warty pig, babirusa, Somali wild ass, Grévy's zebra, Sichuan takin, Chinese goral, Speke's gazelle, Soemerring's gazelle, okapi, red kangaroo, emu, and a wallaby species I've forgotten. Big Cat Country is the other exhibit here, with exhibits that are either too big or too small for jaguar, snow leopard, Amur leopard, cougar, lion, and tiger (Sumatran I think). Lions, tigers, and jaguars are in really big pits while the rest are in netted enclosures that are too small.
Lakeside Crossing: mostly visitor amenities (statues, water features), a sort of breather in the central area of the zoo. Sea Lion Sound is great, a newish exhibit for sea lions and seals. 250,000 gallons in the pool (if memory serves) and a decent land area. The underwater tunnel can get congested if the zoo is busy. Stingrays at Caribbean Cove is a touch pool for stingrays and sharks but it's already closed for the season. Admission is normally $4 for it but again, would be free if you arrived in the first hour.
Historic Hill: a hill with historic buildings

. The herpetarium is nice, and if you like older architecture you'll love it. Mountain chickens (Aka giant ditch frogs), tuatara (if on exhibit), are quite rare, but several other rarities are in the building as well. Outdoors there are exhibits for a few crocodilians and giant tortoises. The primate house has monkeys and lemurs. Spectacled langurs and sifaka are uncommon, and any macaque (if they are still there) is getting to be rare. The exhibitry isn't the greatest. Chain of Lakes is 3-4 exhibits with river otters, Alligator snapping turtles, and waterfowl. The Bird House has a great collection. Prioritize seeing the horned guans- the only specimens on exhibit in America. They're in the central area but may have a see-through curtain drawn for privacy. The exhibits vary in quality, but none are terrible. Outdoors there are aviaries for birds, mostly larger animals like cinerous vulture. The World's Fair Flighr Cage is ancient, but renovations have made it still good. It has North American cypress swamp birds. Peabody Hall usually has an art exhibit.
Discovery Corner: the children's zoo and insectarium. The children's zoo has a generic petting farm, as well as a new Tasmanian devil exhibit, and a building with many odds and ends. There's a slide through another river otter exhibit and a few cages that change their inhabitants frequently. The building has the tree kangaroos and echidnas (I've seen the latter maybe twice), a few aquariums, fennec foxes, parrots, finches, rabbits, guinea pigs, and some terrariums. It's a little hectic but if you want to see those species, then I suggest arriving early so you don't have to pay. The insectarium is good, with many species, and a butterfly walk-through. Also included in Discovery Corner is The Living World, which is in the North Entrance but it changes frequently so I'm not sure what's in there.
Parking: there's a parking lot near both the north entrance and the south entrance, and I think they charge. Otherwise you could park on a street nearby in the park. You'd have to walk, but it's free (just like admission!

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As for what to skip: since the zoo is mostly organized taxonomically, it's up to you. If you aren't a hoofstock fan, skip Red Rocks. If you don't like birds, skip the Bird House, and so on. Have a great visit! Feel free to ask any other questions, and I'll do my best to answer.