Animals are one of the most profitable things you can spend your money on. They may cost too much to start, but they pay themselves back infinitely, because they never die. Below is a list (with pictures) of the animals you can keep in your farm.
I will add a summary of each animal, and proper care later...
Chickens- For proper care hold daily, and make sure you have at least one feeder, and refill it daily or get multiples for multiple days. Chickens produce eggs which are the easiest to gather because you don't have to... they just lay on the ground until you decide you want them. Also if you run out of feed pick up your chicken and put them in a chest until you get more. Summary---easiest animal to keep, less income from produce.
Sheep- For proper care Brush, talk to, and shear daily. Also have plenty of food troughs for your sheep (one per sheep per day) and hay. You have to shear sheep to gain your income, but they also create essence of fluff, which is almost a necessity for winter (I have survived almost 5 years in game without one, one of those was spent farming bats, bees, snakes, and slimes, with the occasional worm to get food.) You can't store sheep like chickens, but you can grow hay almost constantly (not in winter) and harvest every 3 days, plus you don't have to replant if you leave it over winter. Summary---Harder than chickens, but essential if you want to work in winter...
Cows- Cows are pretty much the same as sheep, but instead of shearing you milk them. They also provide the only way of getting water in the winter without a canteen, so they are still somewhat important. They offer a large profit when you get them to produce golden milk, and like sheep you don't need much in the way of food, except to grow hay, which is the easiest thing to grow. Summary---Same as sheep, allows for water in the winter, and substantial profits year round, but if you can't feed 'em... You might lose them (I've never had one die on me... even in that winter I mentioned above).
Hope this helped :)