I've done some training of one sort or another with dogs, cats, parrots, chickens, sheep, rabbits, guinea pigs and fish. I did more trick training and traditional obedience work when I was younger. As I've gotten older, I've moved to training a lot of additional behaviors that just make life easier or nicer. And training to discourage problem behaviors.
I usually just use my voice, rather than a clicker, for my own animals and birds. Clickers are especially good for an animal or bird that's being handled by different people, though. It makes it easier to hand off an animal to another person that the animal hasn't worked with or doesn't have a relationship with.
For my own animals, my voice is the reward, as well as identifying the wanted behavior. It's just personal preference. Plus, I reinforce behaviors all day long, as they happen. I also use a lot of hand signals, along with verbal commands.
I do use food as a reward when calling the pond fish. I should clicker train them. It annoys my husband when they come for me and not for him. I told him he just has to imitate my voice and call, "Come here my lovelies! How are you today?"
I never underestimate a good diversion and a regular routine, though!