I have a bit of experience with this, is one of the things I want to profile myself to. I don’t do it professionally, but I had pretty good results.

1. Eye level. Shoot from Cat’s eye level most of the time, above looks amateurish

2. Have some lights if you can. Two lights and a rim light help a lot. But you don’t need to have lights.

Catchlights are important like in portraits.

Do a backdrop. You can do a small-ish one from a folding table and PVC pipes.

3. Patience. A lot of patience. So I did a friend’s dog, and I was shocked how easy it was. They said sit, and the dog just… sat and posed.

My cats are… well cats. I put them in front of the backdrop. They turn around. One climbed the softbox. The other started running and almost hit the camera.

4. Assistant. This helps a lot. Have someone help you by guiding the cats where to look. It’s hard to do both, but if you do both shoot thorough LCD not viewfinder. Nikons have an option that when you touch LCD to focus and shoot, you can do that and set it to burst.

5. Cans with rice. I used some old 35mm film cans and half-filled with rice. When you wiggle that the kitties will look in that direction, so your assistant can use that. Of course, being cats, they will get bored of it, so you take another can and fill it with coins.

So there’s two aspects, like in human portraits. The technical part (lights, lenses, composition) and the posing part. I will honestly say that it’s harder to get a good picture of a cat than of a human, because, surprise, cat’s don’t pose.

Take lots and lots of photos. I take pics of my fosters for their adoption pages and it takes patience to get good ones. I have my camera available so if the timing is right with light I can try to grab a few if they are calm.

If you don’t have lights or flash make sure they are facing a window but not in direct sunlight when taking the photo. Start trying when they are calm or sleepy, when they are in manic mode you will get nothing. While shooting with one hand, use the other with a toy or something to get their attention to get them to look towards the camera.

A Shallow depth of field is pretty with a nice background to give bokeh.

Many newer cameras have animal eye detection for focusing. See if yours has that and turn it on if it does.

If you make use of a leash (or arm) to help with herding (or branch out into other critters), take a photo without the human or critters in it also to help with removing the human and leash in post.

I’ve mostly photographed dogs, like others have said get down on the ground to their level. Have someone behind you to get the pets attention. Remember they each will have their own personality. My sister has two dogs, one is camera shy, if she sees a camera, she’ll sneak off to another room. The other will roll over onto his back, very submissive. My niece has a dog that will pose when you get the camera out. Crazy! Good luck.

Try to stop down your aperture to maybe f/8.0 or so. You’ll need more light to do this, but pets tend to have longer faces than humans, and it will give you more room for error on focus. It’s a common problem to be focused on the nose instead of the eyes, and on humans this isn’t as noticeable because we don’t have snouts

Hold a feather toy just above your lens to get them to look right at you.

In addition to the pet-specific aspects: Learn the basics of photography first. Practice also on easier, stationary subjects. It’s easier to figure out your framing and such when you don’t have to time an expression or follow motion.

Creative Live, Coursera, LinkedIn Learning among others have good courses. YouTube will require sifting through the people who don’t know what they’re doing. Search terms might include “getting started with photography”, “intro to photography” etc.