Also, I just tried an experiment: With that plump helmet picture I made, I saved it as a .png, a .jpg, an uncompressed .tif, a jpeg compressed .tif, a .tga, and a .gif.
Image quality results:
.png .jpg un. .tif jpeg .tif .tga .gif
High Lower High Lowest Highest Low
Compressed file size:
.xcf (control) .png .jpg un. .tif jpeg .tif .tga .gif
122 kb 48.7 kb 6.01 kb 479 kb 14.3 kb 146 kb 23.7 kb
So yeah, if you want a fast loading image, .jpg's are your best bet. If you want good quality with a medium loading time, .png's are the best. However, if you don't mind only having 256 colors to work with (255 if it's transparent), .gif's are better in quality than .jpg's and don't have much higher of a compression-to-quality ratio.
I tried compressing that image of yours with Paint.net as 8-bit png and gif (7 dithering setting), and the quality is actually pretty good (until you zoom in and see that it's dithered), with png being 21.6 kB and gif being 21.3 kB. The png came to 19.6 kB after running it through an optimiser. JPEG I can vary between a pretty awful-looking 5.7kB at 75% and a near-perfect image the same size as the 8-bit images at 99%. I also got a 44 kB 24-bit png (post-optimising).
So for lossless compression of 8-bit images, optimised png wins. For lossless compression of full-colour images, optimised png wins again. For lossy compression (of an image with lots of gradients), jpeg wins hands-down.