I had to say the DS. The PSP has better controls in my experience, and for downloaded games it is better, but for normal games, the disc is annoying. OTOH, the PSP does have more flexibility for saved game data, and also you can download multiple (full-length) games and put them all on the same memory card. (the games are NOT free, BTW!) A lot of times the DS games that use the touch screen use it in an annoying and/or pointless way (like someone just shoved the touching part in on top of a game that was already complete), but sometimes when the touch screen is really integrated into the play it is a lot of fun. But in general, the DS just also tends to have a better game selection, IMO.
Personally I have never gone to the DSi or the DSiXL or the 3DS, for actual reasons other than money.
The DS Lite has, as Joe 5859 mentioned, a backward compatibility slot for the GBA games. I started with the handheld games with the GBA so it is a concern for me. I am not sure whether it would actually be a concern for anyone buying “from scratch” however, as I have no idea whether it is possible/impossible/cheap/expensive to buy the GBA games now. But that, and the fact that I already have all the cameras and music players I need (OK, several more than I need), and I don’t particularly need to browse the web on my game system, decided me against getting a DSi. Actually when they came out, I bought a new DS, as a backup, because I was (incorrectly) afraid they’d stop selling them. (and also because it was a pretty color. )
My concern against the DSiXL is that a lot of time is spent holding the device. Frequently most of the weight is resting on one or both pinkies. If you play the DS Lite too long and are NOT still 15 years old, your pinkies get sore. The DSiXL is even heavier, so this would be a bigger problem (although perhaps not with a guy, I don’t know about relative pinkie strength. )
Of course, one reason that I haven’t gone to the 3DS is that it hasn’t been released yet, AFAIK. And it is pretty expensive. Plus, the battery life is supposed to be much shorter (according to the Nintendo site) than for the regular DS. I should think it would be quite a while before there were a lot of games available for the 3DS, so mostly you would be playing DS games on it, so it is a lot of money to pay for a feature that won’t probably get much use for a while. Also the 3DS has a motion sensor which I don’t personally want in a hand-held. I don’t want to mess up my game because the plane hit some turbulence.
So I’d probably just go for the DS Lite. But if you find touch-screen controld annoying and like a lot of the downloadable games for the PSP, that might be better for you.
Your mileage may vary.