Get a DS lite! Best handheld I've ever tried, period. Sure, the PSP may have potentially more powerful hardware, but it's software leaves much to be desired and it can't do many things the DS can do to the touch screen. With the "Lite" upgrade, the screens are as good if not better and brighter than the PSP to boot!
The orginal DS may be more durable, see here:
http://ds.ign.com/articles/721/721201p1.html
However, looking at them side by side (and size/style issues aside) there is no way anyone would choose the original DS at this point (unless you are on a budget- you can probably find a used original on the cheap). The screens on the new units are so crytstal clear that it makes it look like you are looking through mud when you view the older one.
I'm lucky so far in that I haven't encountered any problems with my DS Lite (and sold my original). Nintendo has very good customer service, so if a problem should arise, I'm not concerned.
As far as "killer ap" games you should definately check out: Mario Kart, Castlevania, Big Brain Academy, Kirby Canvas Curse (the first one that REALLY showed what the touch control can do) and Metriod Prime hunters. Tetris, Mario and Luigi Super Star Saga, Meteos, and Advance Wars are also very good.