4 devices capable of playing DS games in hardware.
DS (sometimes called a phat DS or original DS)
DS lite
DSi
3ds
DS and DS lite both also allow GBA in hardware. For the DSi or 3ds you are going to want to use emulation, which is not as nice but OK as these things go.
The DSi (and also 3ds) have the ability to play a handful of games in DSi mode if you have suitable DSi hacks (most plain flash carts which work on the DSi just use DS mode) which adds nothing much of great interest really. DSi exclusives are a bit thin on the ground as well. DSiware has a handful of things but the good stuff got ported back to the DS as compilation games. Homebrew for the DSi does exist and does do some more, but still nothing particularly special that you can't readily do on a phone or something today.
As far as DS games go the DSi vs DSi XL or 3ds vs n3ds don't do much different. You have a larger screen on the XL but it is not like that is increased resolution or anything.
The DS lite has a brighter screen than the DS, even hacked versions of the DS (later revisions of original DS could be hacked to have something like the variable brightness of the DS lite). I can entertain a debate either way on which has the better buttons between the DS lite and original DS, however it would be a hard fought one to argue the original DS screen holds up that well. Personally I have used an original DS from the start until today and having fixed a few DS lites I play them from time to time, have not fellt especially compelled to upgrade though.
Nobody ever really sought a DSi or 3ds for the purpose of playing DS games. I don't know if I would go back to a DS lite if I had one of those (other than for GBA games) compared to my definitely grabbing a lit screen GBA to play GB/GBC games in hardware.
No hardware mods worth speaking of like the current stuff we are seeing with the GBA for retrofitting AGS101 screens (a later revision of GBA SP with a nice backlit screen compared to the no light screen of the original and frontlit version that most SPs have). There was an overclock mod that did not do a lot, obviously you can hardware remap buttons and add rapid fire but nothing special. There were a few mods like the DSserial but again nothing terribly interesting that you can not do just as easily with something else today -- you have to remember at the time the GBA and DS represented great little portable computing devices and people played to that, now I can tape an old phone to something or a raspberry pi, an arduino... and get 50 times the functionality for a fraction of the price and power consumption...
There were some cool things with screens like the jumbotron DS which later led to the video capture devices but nothing you can really buy or buy as a kit.
DS hacking is best done by means of a DS flash cart, I would say even still on a 3ds which can sort of squeeze some stuff in there without one. The supercard DStwo is probably the best one out there, a bit more expensive than some other choices (The original DStwo sold out years ago in most places so you are left with the DSTwo+ for $50 USD or so vs the $15 of the cheap and cheerful/bottom of the barrel DS flash cart), however if your requirements are "play damn near every game just as it would have happened on hardware, maybe have some cheats and also be able to play hacked versions of the games" your options become considerably wider. The 3ds does have the ability to dodge some of the flash cart blocks (the DStwo is untroubled by them) if you hack it for it. Note the DS and DS lite never blocked DS slot flash carts, only some methods to launch DS code from the GBA slot. The DSi does have blocks and while there are some new hacks for the DSi it can still be a bit tricky if you don't get that.
Emulation of the DS as an alternative. I can see it happening. Some might question how you get touchscreen stuff to work and for some games it would be a problem, for most a mouse does well enough and it is pretty much going to be the art and some of the really intense action games which want precision input/gestures. I don't know what goes with DS emulation on mobile phones these days but they would be a suitable alternative, might even be able to use a drawing tablet on a PC if you really did want touchscreen. Emulation on the PC also affords to the option to upscale the 3d graphics the DS has which can yield considerable improvements (the 3d hardware of the DS is basic to the say the least).
http://gbatemp.net/threads/hi-resolution-ds-emulation.364549/