Yes you can play games on PDAs. But, their screens are very small. First decide which PDA you want. Basically, they are either Palm based or Win CE based. They are not compatible. Once you know that, you can investigate what games are available.
You can buy detachable keyboards but most games play with the stylus and/or buttons on PDA. Again it will be game/PDA specific
Nowadays, most high end PDAs have nice bright color screens so they are easy to read. But they are not as fast nor do they have the type of storage that desktops do so a game rewritten to play on a PDA may be not as robust as it's desktop counterpart.