A different question from the normal "best flashcart" ones, GBA and DS homebrew have some fairly major differences so I will explain seperately.
GBA
Homebrew is pretty much all the same (in that it will run on any flashcart) with the exception (and rarely used one at that) of the later GBA oriented flash carts which have an extra bank of ram with which to run/store stuff if you so desire (I know of perhaps 4 pieces of homebrew that use the extra ram out of the 1000's around).
There should be other libraries available but this is the only one I have a link on hand for:
http://ezflash.sosuke.com/viewtopic.php?t=150 (EZFlash 3 SDK)
In this case I would suggest reading some of the fine reviews on this very site to get a feel for the GBA carts, that being said I would invest in a DS oriented setup as those do GBA pretty good too.
DS is where it gets interesting: various people and/or the carts makers have released libraries/code allowing people to have direct access (read and write) to the memory cards that the DS oriented setups typically use. That is to say unlike the GBA you can simply tell people to stick files on their memory card and have them read straight off.
Obviously different carts have different access methods so it gets more confusing, the big 4 carts (G6,M3,EZ and supercard) that most people use all have libraries in various states. My apologies to the other carts users (EWin,DS-x, Neoflash and DSLink for I personally have no real info here)
Coding wise they are probably all equal but in terms of homebrew that supports it the M3 is argueably in the lead (of course you can make multiple versions of your code for the carts if you so desire).
For all those with a "normal" GBA cart there are perfectly accepted methods for DS homebrew in the form of a GBA style (one large file) method, in terms of all homebrew this is probably the most popular.
Some homebrew sites would suggest you invest in a GBAMP (GBA movie player) for DS homebrew but if you ever want to run a rom this method is not what I suggest, not to mention it does not have SRAM which means saving is more complex.
As for fast testing GBA wise an emulator like VBA (SDL) or no$GBA are great choices. DS wise emulation is not so great but homebrew can be done there too (some emulators even have support for flash cart libraries).
For fast testing on real hardware go with a memory card using device and be prepared to play musical memory cards.
