advice from me::
1) try to go on a school day/weekday if you can. Holidays and weekends many zoos will fill up with loads of kids and families which can make things a lot harder and a lot more noisy. Though I will say many people do respect someone going to take a shot and will try to keep out of you frame, but you still have to let them past at some point
2) Don't be afraid to use your flash - though I ma guessing that you only have the popup flash on your camera even that can be a help in getting shots if the light is poor - just make sure to have a few folds of white toilet paper held infront of the flash when you use it to diffuse the light (you can use some elastic bands to hold the paper on).
But do watch for the animal reactions when you use flash - some will ignor, others will be disturbed and might try attacking or such - if the animals show distress in any form stop using the flash and revert to normal shooting
3) Use your monopod with your lens definatly - 70-300mm lenses tend not to be the strongest of lenses and can get quite soft at the 300mm end so any support is well advised. (it might also be advisable to try to frame shots in the 70-200/250mm area to try and avoid the long reach softening)
4) To get bars out of shots get your lens as close to the bars as you can and try to keep your lens as straight onto the bars as you can - if you start to change angles the bars can have a greater impact on image quality and AF performance. Also if you find your AF locking onto the bars drop it and shift to manual focusing, slower but more accurate if the AF can't get a lock on to the animal
5) Try to keep your shutter speed as fast as you can - underexposure can be corrected in editing (to a limited extent) and can also be part of a style to a shot - but blur is much harder to work with and cannot be edited out of a shot -- try to avoid motion blur unless you are making it an intended part of the shot
6) set your AF to the middle point only - its generaly the best way to shoot since it lets you define the subject by pointing the camera at it - rather than relying on the camear to choose (because the camera will always pick the closest thing to it since that is how AF functions)