There are many utilities to create animated GIFs from a sequence of individual GIF files. There are also utilities that will produce animated GIFs automatically from a piece of text or a single image.
One problem with this format is the size of the files produced, as they are by definition a sequence of individual images. Apart from minimising the number of frames, the best way to decrease file size is to assist the LZW compression by using blocks of solid colour, avoid dithering, and use fewer colours. If areas of an image don't change from one frame to another, they don't need to be redrawn so make the area a transparent block in the second frame.