The DS lite is far better value than the DSI. The difference isnt really that noticible. DS lite is far > than normal DS. The difference isnt so clear between I and lite.
If you buy UMDs you use them like consoles, insert and play.
I dont think you can install games on the PSP via the UMD.
The PSP GO offeres Digital Distribution, if you plan on having lots and lots of games it might be the way to go, its so much more expensive that i dont see it as i good choice at its current price point.
A few months ago i would have said get the DS. But, recently with many, many, many PSP game announcements i have changed my views. PSP is now of equal value to the DS and it all depends on what games you would like.
Resident Evil
Metal Gear Solid
Gran Turismo
and countless others have ignited my interest in the previous "empty" handheld.
The DS others more classics and proabably a better variety, but PSP others the "hardcore" games as i have mentioned and has graphics.
Hope this has helped answer your questions. If you have any other questions dont hesitate to ask 
