A few thoughts:
1) Understand the animal. The best animal photographers are those who understand their subject. If you understand them you can learn to read their body language, their moods, habits, behaviours. The more of that you learn the more you can not just control them within a situation but also predict their most likely actions.
People say that most animals are unpredictable, but the opposite is actually more true. Animals are as predictable as people, however not everyone spends the time to observe and learn. Read books, talk to experienced owners and trainers. Heck if you get serious take some lessons. Essentially you can learn a lot of it and sure animals will still surprise you, but the more you know the more you'll improve.
2) Learning animal behaviours can tip you off, you'll get a feel for when they'll perform certain actions and also when certain actions will and won't work to motivate them. If working with others pets you can talk to them and pick up clues and learn what to ask about their behaviour. Eg you might find they've got a very hyper dog, but that he always takes a nap at midday; or that their favourite food is tuna etc... Things you can put to use for planning and getting them to respond. A favoured toy can easily get their attention - though most will also build up a collection of whistles, clickers, rattles, shakers, ticktack boxes etc... all to make a little noise. Sometimes something different just gets those eyes and ears focused on you for that split second.
3) In general when working within any set niche of photography the camera is important, but you'll fast find that the best photos come not from understanding the camera better, but the subject (whatever it is). Certainly practice with your own pets, you've already got some ideas of their favoured spots; attitude, mannerisms etc... Maybe you want the cat sleeping in a nice spot on a chair, so perhaps put a blanket in the tumble dryer for a bit to warm it up and lay it down first, lay the cat atop when its normally a sleeping time for them and encourage them to stay - far more likely to work than just grabbing them and trying to make them behave.
As for camera advice
1) Control the scene and lighting as much as you can and remember - dolls/stuffed toys/really chill cats don't care if they just lay there and you take photos for a few hours (though even chill cats will get annoyed if you're using flashes - stuffed toys will sit through it all). Practice on them to get a feel for setting things up - mess around and change settings, see the results; change and play with the angles, the lighting etc... Even people who shoot portraits of people do the very same with dolls and even mannequins - both are ideal to just practice and experiment.
2) Of course if you're going for action shots not just staged or still scenes then things change, its a case of get out there and practice yet again. If you'ev got a pet of your own and someone to help great - take photos when they are having one of their "zoomy/action" moments. Or if you don't find somewhere like local dog agility training/events or equine events etc... Practice makes perfect and the best you can do is, yet again, head down and vary the settings and see the results you get
Myself I only found what worked for me by experimentation. Shooting horses at a showjump event gave me loads to practice on. From that I learned the limits of the shutter speed for the shots I was shooting (1/500sec and I'll get blur on the hooves/tail - 1/640sec and its sharp - so there my lower limit found through experimenting and now its an invaluable bit of info when the light gets challenging as now I know how low I can take the shutterspeed).