The rule of thumb for GIF vs JPG: If you have a picture from a camera it should be displayed on the web as JPG. Line art, text, graphs, or other images where the colors are few and should remain exact should be shown as GIF. JPG file sizes will almost always be smaller than GIF.
As you can see in your picture above, the wires of the cage appear to be very jagged. This is due to the fact that GIF can only support up to 256 colors max -- your graphics application guesses as to what the best color map will be and sticks with that limited group of colors. Here, a few more shades of grey would have smoothed out that jaggedness.
Bitmaps and TIFF files should never be used on the web unless you have a very specific diagram and making the image directly viewable in the browser isn't a requirement (for example, blueprints or archived images like X-rays). PNG files are an advancement over GIF files in that they don't have the 256 color limit, have a more clever progressive loading scheme, and have advanced animation features. Unfortunately a lot of older browsers don't support PNG, but as time grows on PNG will eventually replace GIF.