The best thing to do when resizing images would be to resample when adjusting picture size. I'm sure most if not all graphic applications have it. For Photoshop it appears within the Image size dialog box. There are different flavors of resampling. I use Photoshop and tend to leave the default of Bicubic. That has worked fine for all of the images I have worked with.
When I design any graphics that I may wish to resize, I try to always use vectors and paths as my base design. Flash CS3 has really great tools for this, both in the UI as well as AS3. They always resize well. I just keep a folder with my basic design, pull out what I need and resize it for the pupose of my project. When saving my images for the web, I use the same helpful hints and steps as outlined above. ;)