It depends on the nature of the image, if it's a photograph, JPEG would give the best compression/quality relation, if it was pixel stuff like writing or pixel images, or have some transparency, then it's GIF/PNG to choose (GIF,PNG8 offers one level transparency, while PNG24 offers a leveled transparency foreach pixel).
When I'm confused, I usually save the picture in all three formats and decide what gives the best quality/size deal within the size limits I need, I also try to lower down JPEG quality to the level where the image is still good quality (cuz that varies from image to another).
Also, if it was a photograph with some writing, split it into a JPEG photograph with a transparent GIF writing overlay (cuz writing edges look distorted in JPEG).
So, when you are confused, give the three a try and decide, with time you'll gain experience what format suits what content the most.