Often enough I find myself flipping through a book, or just browsing around websites, and I'll come across those simple lists that say something like 'birds are associated with this, cats with this, dogs with this' all without ever explaining why these animals hold such associations.
Now, sometimes such associations are fairly obvious. For example, a lot of people associate caterpillars/butterflies/moths with transformation, of course because of the physical transformation they all go through.
Other associations might not be so obvious at first glance. When you think of a vulture, what comes to mind? For many they hold associations of death and decay, as they are scavenger birds. However, for the ancient Egyptians, they also noticed that vultures made loving mothers - and so you often see their mother Goddesses, such as Mut and Aset (Isis), wearing beautiful vulture headdresses. For this reason, if you're trying to understand the specific symbolism of a particular culture, you'll have to try to view the animal from their position. How one culture views an animal might be very different from another.
Generally speaking though? If you want to learn the symbolism and associations of an animal, try going right to the source. While individual animals often have their own personalities and quirks,
when we look at the whole common themes often stand out. Read up on the animal. Watch it in person if you can, and if not, look around online for videos. What does this animal look like, where does it live, what does it eat, how does it act, what are it's habits, is it social, what are the breeding habits, how does it raise it's young (if it does), what does it sound like? All those sorts of things.
Then consider other sources. Look at folk lore and mythology. After learning about the animal, you might pick up on symbolism that you would otherwise have missed. See how others view the animal, and consider why they have those views.
In many cases, symbolism is very personal. Especially when we look to symbolism in dreams, found in meditations, or symbolism that we may use in ritual. If you come across symbolism that doesn't speak to you, even after giving it a good bit of thought, then don't feel you have to use it. Figure out what makes sense to you, what works for you, and go with it.