Bit late to the party, but we're zoo members (family with 1x 8 year old girl, 2x 4 year old boys) and go pretty regularly ... probably 6-8 times a year. But in case it helps anyone researching ...
You're going for one day, and it's a 'first ever visit' or something similar - a rare trip anyway. You don't have a particular list of animals to look at.
The thing is, Chester zoo is WAY too big to do in a single day. Don't bother trying, you won't enjoy it. There's some enclosures which are great, but take up a load of time for one reason or another. There's others where they're good, but your chances of seeing an animal are low, because they're just not that sociable. The route I'd probably take is:
1. Come in through the entrance (obvs), go right. Follow the path round and you should see the Black Rhinos. You can quite often see them in the indoor part of the enclosure, they're magnificent animals.
2. Go through the Painted Dogs enclosure. They're quite cute when they're all sleeping together.
3. From there, go past the roan antelope and through the Islands section. You'll see the Provosts Squirrels, Visayan Warty Pigs, Tree Kangaroo, and Cassowary. If you're lucky, you might see the Sumatran Tiger.
4. Keep following that path, and you'll see the Sun Bears and Sumatran Orangutan. I'm going to be controversial and say that whilst it's ace and definitely worth seeing, give the Monsoon Forest a miss. It'll take up too much time on this schedule ...
6. Also give the lazy river a miss, you won't see any new animals on that. If you need play time or food, the Manado Street Kitchen is a good place to sit. Take a picnic, the food prices are rather high. You'll see Macaques and Babirusa on your way round.
7. From there, follow the path on the right past the Capybara, Spectacled Bears, and Tapir. If the Fruit Bat Forest is open, go in.
8. Past the fruit bat forest, take the first right over the wooden walkway. There's sometimes animals there, sometimes not. On the far side, you'll see the Cheetahs.
9. The eastern side of the zoo is really good & worth a visit - here we're just taking a whistlestop tour around it however. Go past the June's Pavillion restaurant , then go take a look at the Giant Otters. Make sure you go to the far end of the enclosure, there's a great underwater viewing space there.
10. Take a left when you get past the giant otter house, walk past the indoor part of the penguin enclosure (don't worry, we'll come back to them!), take a walk through the aviary with the Flamingos in.
11. Take a left when you get out of there, then left again so that you can see the Humboldt's Penguins. The enclosure's great.
12. Walk down the side of the flamingo enclosure. At the bottom of that path is the Chimpanzee enclosure. Take a left turn there.
13. Walk down that path. You might see some Buffalo if they're out, and catch a glimpse of the giraffes. Walk past them, however, and past the toilets. Take a right when you've passed the toilets, over the bridge.
14. You'll see a picnic lawn in front of you. Walk past that. If you have time, the Butterfly house on your left is good. If it's getting late, walk past it. You'll get a good view of the Giraffes on your right hand side The Lion enclosure is at the bottom of the path, and worth a look.
15. Come back from the lion enclosure, back past the giraffes and butterfly house. After the picnic lawn, turn right. You'll see the Camels then the One Horned Rhinos on your left, and the Elephants on your right. If you have time, the Monkey house is on your left after you've passed the elephants.
16. The exit is on your right. You will be exhausted at this point.
Note the above itinerary is quite heavy, You're going to need to turn up fairly early (10am-11am) and stay fairly late (~4pm-5pm). But you'll see almost all the big ticket animals at the zoo, and see some of the nicest enclosures. If you can't make it for that long, then shorten the list above rather than try to cram it all in.
Possible Alterations
You might want to see a particular animal that's not on the list. If you want to see more of the western side of the zoo .... instead go Entrance -> Monkey House -> One Horned Rhino -> Fruit bat Forest, then pick up the above itinerary at #7. This will cut a load of time off your visit, and give you more time to see things like the Lemurs, Komodo Dragon, Jaguars, etc
If you want to go to the Monsoon Forest, which contains amongst other things the Crocodiles and lots of tropical birds then I'd follow the above up to #8, then go straight from the cheetahs, walk past the penguins (skipping the giant otters), walk by the flamingos but not in to the aviary, and pick the route back up again at #11 (Penguins). You may well have to drop #14 (Lions).