The zoo is a great place to take photos of animals up close without them being able to run away or eat you. You can be within a metre of them so you might think your 17-50mm kit lens will be OK. Unfortunately, such a wide angle means you will have a lot of the background in shot like the fences or bars and possibly even other zoo goers. Even if you get a photo of an animal it will be tiny in the photo. Seeing so many people trying to take photos with mobile phones at the zoo makes me cry! With the right kit, you should be able to take a photo that could convince somebody you were on safari instead of in a city centre.
Clever zooming and cropping means you can cut out the surroundings with a 400mm lens.
Go large: you should try to take the longest lens you have, and if you have a 2.8 or lower f stop lens then it's even better. Most zoos don't allow flash with good reason so you need all the light you can get from what is available. There are a lot of unusual animals in dark or dimly lit exhibits that are out of the range of most kit lenses. Use the best lens you have, increase the ISO, use a tripod or brace your camera against something and you should be able to get a decent photo. Just make sure you are not in an auto mode which will make the flash fire and get you in trouble.
This cutey was shot with a Canon 100mm macro without stabilisation at f2.8 in a near pitch black exhibit. I used the wall next to the glass to keep the camera still.
Animals often follow certain behaviours and have a daily routine. But if somebody bangs on the glass, you can expect them not to settle down for a while. Watch what the animals do and see if they make any repeated behaviour that will help you predict the time for the best photo. Choose your moment, have the camera ready and looking between the bars or close in to any glass to avoid reflections, be still and patient and you can get some photos at the zoo to be proud of!
Wait for the crowds to pass and the animals will behave more naturally.