According to NASA, animals adapt very quickly to weightlessness, and mostly behave how they normally would. Because they are weightless, there are some complications that the animals have to deal with. For example, fish that normally swim in a line may swim in circles because they cannot tell which way is up or down. NASA also reports that small mammals may have a hard time because they normally curl up and cuddle for warmth, but it is hard to do this when an animal is weightless. Here is a longer article from NASA about animals in space:

It depends on the animal - animals that normally live in water behave as though they are in water, which makes sense because animal body density is very close to that of water, and as a consequence animals are very close to weightless while in water.

Other animals get rather seriously disoriented.