The bmp file does not use any compression to store the file and will always produce the largest picture when stored on your hard drive. The Tiff or Tagged Image File Format produces a nice image quality and comes with several different compression's available when storing the image. This is a nice format to use. Jpg images also use a strong compression. This format will generally produce the smallest image, however you must be careful.
Jpg compression uses an algorithm that identifies trends or estimating of colors.
So if you use Jpg, don't use it as the file that you store your permanent photo with. Each time you compress and uncompress the image for editing, you will lose image quality due to the estimating. Store you file in a bmp or tiff and convert it to a jpg just to store it on the internet. This is a great format for the internet. Gif images were developed by CompuServe and are designed to hold a series of pictures for animation.