Can you plug your PSP into your TV and play games?

Yes you can. You have to either modify your FAT (PSP-1000) the older larger one, by taking it apart and adding some hardware, which is rather technical.

Or simply by the plug for the SLIM (PSP-2000) and hook it up to your tv via the composite or RCA\S-video plugs. Make sure your TV has these plugs, and get the appropriate cable that will work with your tv! MUST be NTSC. You can also use a USB cable and plug in the PSP to the PS3.