It's called morphing and warping. And no, it's not going to be something quick and easy you can learn just by watching a tutorial, as you are treading onto the dangerous area of character animation and its subtleties. Do some generic research on the matter and check making-ofs of talking animal movies like Dr. Dolittle or Cats&Dogs to get an idea what you may be getting yourself into....
In the olden days, we'd rub some peanut butter in the animal's mouth and dub in the audio.
This is not a task I would approach in After Effects because, as Myl states, it's all about characterization these days. AE can composite your 3D character into your clean plate background and help you do spot corrections but creating a talking dog is nearly impossible.
However, creating an animated talking dog from artwork, that's what AE was made for. Using audio tools and expressions you can make a mouth element move up and down and scale in response to frequencies and amplitudes. You can use a tongue layer to add credibility. Or you can create a library of phonemes that you can switch in and out of a comp in response to the track using an expression controller. Classic animation requires about a dozen mouthshapes and, at 12 frames per second, making a cartoon of dog speak isn't particularly difficult. But AE is just the tool. Designing the dog and studying mouth animation are required first.