I know few people who have different kinds of animals and do some farming, although only one really is somewhat self-sufficient in food, others do it as additional source.
Goats are hard to start. They are overly smart and bold, they open locks and easily go through any fences you have. Experience in handling is needed. Pigs are easier to handle and give pretty good amount of meat, but they are costly and also need much food. Cows are good only for those who already have everything, they are too large.
Best animals are rabbits and birds. Rabbits easily die from diseases and require proper handling (for example, they may try to even eat each other in tight space). But they are compact and require less space - you may place them in small room with cages, you don't even need to get plot of land for them. They also multiplying fast.
Birds (chickens, ducks, geese) are easy to handle. Better to get muscovy duck or turkey, they are gaining weight very fast and undemanding to conditions. Geese are same in terms of conditions (they can easily sleep outside with -15C and snow), but they have bad behavior - they are goats of bird world. They also may try to fly away at autumn.
So, in terms of livestock, I'd recommend starting with rabbits of birds, and then go into heavy category.