I have PSP 7 and here are the steps I usually take.
1. Load the image.
2. (in the menu) Image > Resize
3. (in the resize window)
Resize type: Smart size
Resize all layers - checked
Maintain aspect ratio - checked
Then I generally have Pixel size selected and either fill in the desired width or desired height. By having maintain aspect ratio checked, when you change width or height, the other is adjusted proportionately.
4. (in the menu) Effects > Sharpen > Unsharp Mask
5. (in the Unsharp Mask window)
I generally start with the following settings:
Radius - .50
Strength - 100
Clipping - 5
I'll sometimes adjust Strength to see what I like. Somewhere in the range of 50-150/175ish usually does the trick.
6. (in the menu) File > Export > JPEG Optimizer
Yes, for some images, GIF Optimizer might work better. If you need to do transparency, then you have to make it a GIF. However it's easiest to explain as JPEG.
7. (in the JPEG Optimizer window)
Compression - a setting of 10 is a good starting point. Compressing more than 15, 20 will definitely make the image smaller, but around that point the quality begins to go downhill. Play around and save a couple and compare. See what you're happy with.
Format - I usually go with what gives the smaller size.
8. Then you'll be prompted where to save it and then you're done.