I agree...quail and rabbits are pretty easy to deal with and are easily moved if you move. Chickens and ducks are harder to move...goats too. You can't just have one goat they will scream their bloody heads off if lonely.
First find out what your landlord will accept and then you will know what to look for when checking out the city/county ordinances. Quail and rabbits can be raised in small moveable cages and will probably go unnoticed by neighbors cause they don't really make any noise to speak of. If you can take care of a cat/dog you can probably take care of a rabbit.
If your landlord won't let you put in a garden then do some research on container gardening if you are interested in growing some of your own food. Start with stuff that is pretty foolproof like cucumbers, cherry tomatoes, bush beans, lettuce, and spinach. They can easily be grown in containers and moved if you move.
If he won't go for that then I suggest you find out when and where your local farmer's markets are. See if there is anyone local that will sell you milk, either cow or goat, and learn how to make yogurt and cheeses. Start by learning how to preserve your own food, buy in bulk from local farmers and put up enough food to get you through a specific time frame...say, 4-6 months. Learn how to can or freeze or dry food for later use. See if there are local sources for meat that you could trade services/products for or offer to help with processing in exchange for meat or a discount. That way you learn a skill and get something out of it as well. Make friends with local farmers...it can only help in the long run.
Learn how to sew and do clothing repairs yourself instead of throwing things out. Make your own detergent/soap. Start researching different projects so that if you ever end up in a place of your own someday, you'll know what to look for based on what projects you want to do.
All of these things are steps towards being more self sufficient. It is a process and if you tackle too much at once then you may set yourself up for failure and give up completely. So, focus on one or two things right now that you can learn and move forward with it.