Find a good and free program on the Net. An excellent and free fractal program is Sterling2, available
Nearly all fractal generating programs give the user one or more algorithms, the capability to zoom in and to save images to disk. So what distinguishes an excellent program from an average one? Consider the following criteria, roughly in order of importance:
1) The richness and complexity of the images one can produce with the program. This applies to two major aspects of the images: a) form and b) colour.
a) Richness of form. The program should enable you to produce complex patterns with rich detail. Above all, it should let you create images that have complex and interesting structure, as opposed to random-looking blotches. (Hands up all those bored with the familiar Mandelbrot split circle.)
b) Richness of colour. It should use 24-bit colour (also called 'true colour') rather than 256 colours. 24-bit colour allows one to generate up to 16 million different colours.
The program should create images whose colour varies continuously rather than discretely. It should produce a variety of colours and textures, as opposed to solid blocks of colour, dotty images, or silhouettes.
2) It should offer a variety of algorithms giving different types of fractals. The program should come with a fair number of built-in algorithms, or else allow you to specify your own. If y = z*z is the only available equation, even though this gives the incredibly rich Mandelbrot set, we are likely to lose interest before long.
3) The scope it gives for manipulating the images produced. This includes colour alteration, the use of filters to produce different effects (perhaps completely changing how the fractal looks) and other ways of manipulating the image.
4) Utility. It should be easy to use, intuitive and fast. 'Easy to use' is hard to define since whatever we are used to appears easy and natural. When we change to another program with a different user interface then this new program may initially appear 'difficult' to us. Thus a program that seems intuitive and easy to use to me may seem difficult to you, and vice versa. Also: is useful help available?
5) Cost. Unless the program is streets ahead of the free fractal generators available, there seems little point in forking out money for it, though you may be willing to put up with being nagged to register. There are many high-quality freeware programs. You can try doing a search for 'fractal' at a freeware software listing or just google "fractal programs".
Download and install a fractal program of your choice. Run it.
Choose a formula and draw a fractal. This is the coarse tuning.
Zoom into an interesting or promising part of the image. Repeat until you get an image you like. Draw the image so that it is as large as your screen resolution, eg 1280 x 1024.
Get just the right framing of the image.
Adjust the parameters available in the program that control how the image is drawn, such as filters, renders, transforms or whatever they are called. Also, try the Julia version. This is the medium tuning.
Adjust the colours using the tools provided by the fractal program. This is the fine tuning.
If the image has fine detail then the next step is to perform anti-aliasing. Anti-aliasing smooths out the jagged diagonals and makes the fine detail look more coherent. It makes curves smoother by averaging out their trajectories and does dithering, ie interpolates intermediate shades where there may only be sharply distinct colours in the original image.
Draw the image larger in each dimension by a factor of 4, eg draw the image as 5120 x 4096 pixels if you want the final image to be 1280 x 1024 pixels. Naturally, this will take 16 times as long as rendering the normal-size image. Once the over-size image has been rendered you need to bring it down to your desired resolution.
Some programs, such as Sterling and Sterling2, have a feature that does this for you. In Sterling do Image -> Anti-Alias 4:1 and save the resulting image. You can do the same thing in any good graphics program, such as Photoshop, where you just hit control-alt-i. Other programs that will let you resize are ACDSee and Gimp.
Some people sharpen the image after anti-aliasing, as it can look a little soft, but this is a matter of taste.
You may find that the new image is much smoother and less speckled than the original. Do give this a try at least once, as you may be surprised by the difference.
Cycle through the colour gamut using a graphical program. The image produced by your fractal generating program will be coloured in a random way from the point of view of human esthetics. The colours may be bold or dull, they may look harmonised or not, they may not be to your taste. Almost every fractal program allows you to alter the colours of the fractal produced. However, the colour controls tend to give unpredictable results and require you to render the fractal anew. This takes time, which is always a scarce commodity. An alternative method that is much faster and also more intuitive is to do post-processing in a graphical program, such as Photoshop or ACDSee.
You can cycle through the possible colours by gradually altering the hues in the entire image. To do this in Photoshop, use Image -> Adjustments -> Hue/Saturation and move the hue slider through every value, both right and left of the centre. Save the images that appeal to you. Then hit control-I to reverse the colour values and repeat the process.
The idea is twofold: to obtain better colour combinations as well as to highlight different aspects of the fractal's structure. Not only can a dull-looking fractal become striking, but new structure can be revealed. It will give you ideas for what to zoom into - areas that suddenly become interesting after colours or light/dark have been changed.
For best results persevere until you hit something really interesting and then explore it. The easiest way to make a good new fractal is to continue exploring one of the best ones you have made previously.