Online Marketplaces
Online marketplaces such as Amazon, eBay and Half have a number of older PlayStation titles up for grabs. In some cases---if you're willing to spend a little extra---you can even find certain titles new, sealed in the original packaging. It will undoubtedly take a bit of persistence, but it's possible to score some great finds through an online marketplace.
PlayStation Store
If you own a PlayStation 3 or Sony PSP (PlayStation Portable) console, you already have access to a number of "PSone Classics" for direct download. By signing up for the PlayStation Network, you can access the PlayStation Store through your console and purchase a number of original PSone games ranging from roughly $6 to $10. A credit card may be required to add funds to your PlayStation Store wallet.
Third-Party Sellers
Much like online marketplaces, third-party sellers in forums are an alternative to downloading games from Sony or buying them directly from websites like Amazon. Certain gaming forums, like NeoGAF and the forums on gaming blogs like Destructoid and IGN, may have threads (or conversations) where people can buy, sell and trade games. Buy from third-party sellers at your own discretion and never give out any personal information to a stranger (like credit card numbers).