Disclaimer: The information regarding hacking a Nintendo DS or Playstion PSP is intended to run homebrew apps, which are 3rd party software and not official and in such is no way or form consenting or encouraging the usage of illegal obtained ROMs.
it depends on what youre looking for. DS is plain out for gaming but the PSP can run movie, music and other media. both systems are incredibly easy to run homebrew on. the DS/DSi only needs a simple flash cart.
i HIGHLY recommend getting either an acekard 2i ($13) which works for the DS and DSi or cycloDS ($55). i chose the acekard because of its price; the cycloDS has a couple of extra features like live save and such but doesnt warrant the extra $40. both communities are very active in updating firmware. if you chose the acekard, load AKAIO onto and you're golden.
the difference between the DS and DSi are substantial. the DSi has an integrated cam, faster processor, bigger screen and supports WPA where as the DS is WEP wifi. however, it gets rid of the GBA cart and costs $50-60 more than the original lite version. check gbatemp for more info.
the PSP is also very easy to run homebrew. all you need is someone with a modded PSP and you can load custom firmware on to. or you can buy the components. you need a magicstick pro duo with jigkick loaded and a pandora battery. one thing to note is, there are different versions that have different hardward such as mobo. thusly, some will be more difficult to mod. check psp-scene for details.
lastly, all these parts can be purchased from dealextreme. they are a company that is based in HK so the parts TAKE FOREVER, like 3-4 weeks, but they cost a lot less than US based companies and they always deliver. just be patient.
PS i have and know a lot of people who have both systems and byfar the DS gets played more often than the PSP.