You can get some pretty good images out of almost any image editing software, even GIMP. First, know which format to use, and size your image appropriately.

For JPEGs: use chroma downsampling, and don't go above 80% quality unless absolutely necessary. If your software has a preview function, use it. Encode with MozJPEG if you can.

For PNGs: compress the hell out of them. There's several tools that can compress them further.