Good choice on the DS - It's a beauty of a console
Homebrew for the DS more than certainly exists - I've made a couple games for mine already.
Homebrew for the PSP also exists... Infact, the Homebrew devkits that are available are both part of the same package -
devkitpro
The DS has Internet abilities via the Opera browser, which will be released as a game cart.
There are over 400 games for the DS (each game has a unique game-code, but some games, like Metroid and Mario 64 double up, so take the 400 as a guesstimate), not including all the really fun ones available via the homebrew scene.
If you want to get into deving for the DS, the hardware that you get will mean you can also dev for the GBA (both systems use an ARM7, and the DS has an extra ARM9 as its main processor).
I like to play games on my consoles, and as such, I bought a DS.
When I got into Homebrew for it, I realized I could then use it for other things to, by using free homebrew applications available for free, and I already had the hardware to use them.
The graphics capability, and processing capability has nothing on the PSP, but remember the good old days when we only had a few hundred K of RAM? And the games then... The funnest ever! Who says you need DOOM3? (I sure don't!).
Another dodgy thing about the lite... The old GBA cartirdges stick out about half way from the bottom since they made the slot smaller.
Made quite a lot of people p!ssed off...
Personally, I have one of the original versions, so I don't have that problem - But, if you're buying homebrew hardware for the DS, keep it in mind that it will stick out!
Anyhoo... I'm very, very glad with my DS purchase, and am happy that I also got into Homebrew for it also
Good luck with yours in the future!