This isn't a tutorial. The truth is that you won't learn much about making fractals until you have a fractal tool installed and you're messing around with it. This is an introduction, a collection of useful concepts, search terms, and links that will help you take that first big step into the wonderful world of fractal rendering.

For starters, here's a timelapse video of me designing a fractal "from scratch" in my preferred tool. This is a great look at what it "looks like", but it's not really instructive. I have a real-time video with some basic info, but it's much longer. I work primarily with "flame" fractals, which use the FLAM3 algorithm published in the 90's. If you ever used the Electric Sheep screensaver, you saw some delightful flame fractal loops. If you want to learn more about the math behind fractal rendering and complex numbers, this interactive guide on How to Fold a Julia Fractal is one of the best pages on the entire internet.

The best way I know to explain how these tools make the interesting patterns is with this Numberphile video about Chaos Games. The process used to draw the Sierpinski Triangle is the same basic process that flame fractals use. The video makes it easier to understand, but the algorithm iterates between various algebraic transformations in a chaotic order. Picking the right functions to transform the image with, and defining rules to the chaotic order that they are applied in, fractal artists make patterns that can be immensely complex, surprisingly expressive, and stunningly beautiful.

You don't actually need to do any calculations to make fractal art. As you can see in the timelapse video, I'm usually adjusting various parameters just to find out what happens visually. Understanding the theory helps make sense of the tools, but you can learn "by feel" just fine if you can get over the hump of learning the interface.
You also don't need a fancy computer, but you do need a computer, not just a phone. Lots of artist render without even using their GPU.

If this sounds exciting to you, I would suggest you use a different tool to get started, called Chaotica. There's a bunch of different reasons, but the main thing is that Chaotica has more developed community resources, and is still (sometimes) in active development. And for getting started in Chaotica, there's nothing better than Pugnacious One's Friendly Tutorial Collection. A lot of effort goes into making those guides accurate and easy to follow, and there's plenty there to get you started.