Taking pictures of your pets can be challenging, I know. when caught at the right moment, they all look like angels, we all know that it can take a while and a little bit of work to get that perfect moment, so be patient. As a professional photographer, I have all sorts of lights and flashes but not to worry about that as I find it way better to use what we call a continuous source of light. The cheapest source of continuous light you can get every day is the Sun. You can achieve great results just by photographing your pet next to the biggest window you have (Try to avoid direct sunlight).
In my case, I use a studio light, the Godox SL 60w, with a softbox, which helps me to shape the light with more precision. As for the camera and lens, the following pictures were taken with The Fujifilm XT4 and XT3 pair with an 18-55mm lens. Even if I prefer the quality of a prime lens, I find it way more comfortable to work with zoom for pet Photography.