I'm not the right person to create comprehensive tutorials, but there are a lot of sites and homepages that have this information written in a better way than I could do. This part of the fractal pages will be a collection of links to tutorials on several fractal softwares.
As a starting point, I can say that I've tried many different fractal softwares, and it's impossible to say which is the best or which is the worst. They all always have one thing that isn't available in the others, which make them unique to obtain certain kind of effects/images. Some are more user-friendly, some require you a bit of knowledge about fractals and how they "behave" to get you started. Some aren't exactly fractal generating softwares, but some of their painting/drawing techniques use mathematical formulas to make the images, therefore they are a bit into the fractal realm. In this link, you can find a list of several fractal generating sofwares, choose yours. It's not a complete list though, there are many others.
I started around 1997 using Fractint (in DOS), which didn't have the ability as the new softwares have like true colour, ability to save your images in big sizes, print capabilities, and so on. Well in fact Fractint can do all these things, but it requires you to "tweak" the software a bit using some scripts and commands to make what you want. And not always the results are interesting. But I still use Fractint, as a starting point for more elaborated images after importing them into Ultrafractal or as a stand-alone creations. It's old, but it still satisfies me. One tip I could give to new fractalists is to never give up upon a blank screen that seems that will never change to anything interesting. Start trying different things, search for help files on the formulas you're using, and so on. It's easier than it seems. OK, enough talking.
What is the Mandelbrot set? by Fabio Cesari - a mathematical and visual explanation about the Mandelbrot set.
What is the difference between Julia sets and a Mandelbrot set? by Fabio Cesari - explaining the differences between the 2 most common kinds of fractals.
Ultrafractal Jump Start tutorial by Janet Parke - for those who don't know where to start in UF.
Tiera-Zon Tutorials - Tutorials for the fractal generator Tiera-Zon, you can download it here.