The highest version of iOS for a 1st gen iPod touch is 3.1.3, which appears to be what you have (based on your profile info).
You can still download apps that work with 3.1.3, but it is difficult to find them. Most are older apps. Current apps usually require a more recent version of iOS (or more recent hardware).
I have a 1st gen iPhone, which is the same as a 1st gen iPod touch for iOS version. I recently found a helpful app, called VintApps. You can find it in the App Store by searching for that name. It's not free, but the price is 99 cents. What it does is "filter" the content of the App Store based on hardware and iOS limitations (that you set), and then present a list of apps by category that are supported.
What I do is find something that looks interesting using VintApps. Then, I use the App Store, in iTunes on my computer, to find that app. I use the App Store on my computer, because it is much faster and the screen is much larger. But you can link directly from VintApps to the App Store on the device.
In addition to VintApps (mentioned above) you can try iOSSearch.
Select your iPod model from the menu and type whatever you want in the search box. It filters compatible apps and games in Apple's iTunes store.