There are a number of methods for reducing filesize on images. Resizing it to a smaller image area is one method. Changing the resolution of the image to a lower resolution is another method. Selecting a file format that has some built in compression like (JPG/JPEG) is yet another method. From there, your creativity can take over.
If the image is something without need of a bazillion colors, reduce the color resolution from millions to (32-bit or 64-bit color) to hundreds or fewer (16-bit and 8-bit). This will decrease the image quality, but as you shrink images the need for millions of colors starts to go away.
Realize, that there is rarely a need to do all of these things. I typically resize my images to meet new size (as in area XXX by YYY) limitation and then reduce the resolution as necessary to get the final filesize down memory-wise.
I use a share ware program called Graphic Converter (by Lemke Software). It looks to be a Mac specific bit of software, but it handles everything I need.