Surely. Firstly, it looks like Progressive Scan doesn’t work through composite (which you have right now), so don’t worry about that quite yet.
The first thing I would do is get a Component Video cable. You can get one at a Gamestop or any electronics store for about $25 if you don’t want to wait, or if you can wait 4-5 days, get this one for $4.50 plus postage.
With the three cables you have now, the yellow one carries video, and the red and white carry stereo audio. With that component cable, the red and white pair will still carry stereo audio, but you’ll now have three video cables, one green, one blue, and one red. This will improve your picture quality greatly.
After you get that cable hooked up, you can turn on Progressive scan in the settings menu of games that support it. Unfortunately, not all games support it. Here’s a list of games that do. The first column, 480p, shows games that support progressive scan. The second column shows games that can be switched into 16:9 aspect ratio, meaning they won’t be stretched or letterboxed on your TV. It also shows that Gran Turismo 4 apparently supports 1080i resolution, which I didn’t know the PS2 was capable of. However, keep in mind that the difference between Progressive and Interlaced won’t be nearly as obvious as the difference between the two video cables.