Aside from simply getting outside and watching animals (the best way to start!) there are several online resources where you can explore animal behavior:
I had the great privilege of spending the majority of my time in graduate school at the Macaulay Library at the Cornell Lab of Ornithology. It is the home of a giant archive of animal behavior media, from sound recordings to videos. It has a searchable online database, where you can find recordings of animals from around the world.
Another resource for bird songs is xeno-canto, which also has a searchable database.
Want to help scientists? There are many citizen science projects where you can help scientists gather information about the animals around us.
Here are just a few:
From the Cornell Lab of Ornithology:
Celebrate Urban Birds investigates birds found in urban areas.
NestWatch participants follow nests as birds raise their young.
Project Feederwatch participants count the birds that visit their feeders and help scientists keep track of bird populations across North America.
Firefly Watch (with MassAudubon) tracks firefly populations across North America.
For more citizen science projects, Scientific American has compiled a listing on its website.