A long zoom lens will be handy as you'll be able to get close to the animals without having to climb into the enclosures. Something around the 70-300mm mark or bigger would be good. Also, consider taking a macro lens along as most zoos have enclosures where you can get close to insects.
A camera with a tilting LCD screen is perfect for zoo photography and you could take a monopod along to raise your camera up above the fences but leave your tripod at home as they don't mix well with crowds.
Pack a brolly as it will most likely rain at some point during your visit and have a lens cloth handy to wipe off raindrops that will blur your shot. As you have limited angles to work with you may have to shoot into the sun so a lens hood would be handy.
A polarising filter will be good when you're shooting through glass as it reduces reflections it will also reduce the amount of bounced light so the textures and tones in fur will stand out.
Pay attention to the weather forecast. When it's raining you'll get drenched kit and most animals will head indoors where you can take photographs but you'll have glass and crowds in a small space to contend with. If it's gloriously sunny is may be too bright and you'll get very harsh shadows. You can use fill-in flash but check before you do as it's often not allowed. You're better off sticking with natural light and increasing the ISO instead. Rain's too wet and sun's too bright but an overcast day's just right. A slight covering of cloud acts like a softbox so you'll have images that have even tones and are well balanced.
Plan And Research
Before you set off, go on the zoo's website, find a map and make a plan. Arrive early to beat the rush and try walking round the opposite way to the crowds to give yourself chance to capture shots without the crush. Feeding times are great photographic opportunities but they're popular with visitors so arrive early.
Cages And Glass
Unfortunately zoos are full of cages and there's nothing worse than shooting through wires and bars! Sometimes the gaps are just big enough to poke your lens through but if they're not, get as close to the fence as possible, position your lens so it's pointing through one of the gaps or, when the fence has small gaps, make sure that the face of the animal you're photographing is in a gap, use a wider aperture setting and wait for the animal to move back from the cage. This way the fence will be thrown our of focus so you, hopefully, won't even notice it. If you venture indoors you won't have fencers to contend with but glass full of greasy smudges will certainly be in your way. To minimise reflections attach a lens hood or hold your hand to the side or above the lens. If there's a lot of people touching the glass switch to a slower shutter speed to minimise shake. You may also need to switch to manual focus as cameras can be fooled by glass.
Find Good Shooting Spots & Angles
Make sure you take a walk around the edge of the enclosure before you take your photos to find shooting locations that won't leave your image with a distracting background or posts sticking out of the animal's head. Try to avoid shooting down as this can distort features instead get down low, to eye level if possible, to create a more dynamic shot. Use a wide lens setting and crop in later to make sure you don't amputate any limbs by accident – a shot of a monkey missing its tail is very can be very annoying to look at. Don't be afraid to fill the frame with your subject as this will give your shot more impact and it won't be so obvious that you took your photo at a zoo.
Focus And Shutter Speeds
Most of the animals won't stay still so use focus lock to prefocus on a certain point and take the shot as the animal enters the zone that's focused. Always focus on the eye and try using continuous shooting mode if you don't manage to get your subject in frame the first time. Try freezing their movement with a fast shutter speed and if you're panning, use a speed between 1/8sec to 1/30sec to blur the background but leave the animal sharp.
Keep an eye on your white balance when going from indoor and outdoor enclosures and watch out for condensation when moving from the cooler outdoors into the tropical climate of a butterfly house. You'll need to give your camera time to acclimatise otherwise you'll end up with hazy, dream-like shots.