You just need a flashcart, a slot 1 cart, and a micro SD card. Most of the new slot 1 flash cart support the bigger capacity microSDHC cards so you can one of those.
If you ever plan on buying a DSi in the future, you might want to buy a slot 1 flashcart that supports DSi. Most of the new flashcarts support both, and have an "i" in there name, like the EZ Flash Vi, Acekard2i, M3iZero, etc. Of course these carts also work on regular "phat" and "lite" DS models.
Using the flashcart usually just means formatting your micro SD cart, then putting your flashcart's firmware on the microSD cart, putting some NDS roms on it and then sticking the micro SD card into your flashcart and then stick the flashcart into your SLOT 1 slot (where NDS cartridges go). And that's it. Much easier than the PSP world where you have to hack the firmware and stuff.
Whatever flashcart you buy, just consult the FAQs/Wiki/Sticky threads for that card in the various forums here.
As for stuff you can do besides playing NDS roms, you can run homebrew app/games, run emulators, etc. See the homebrew NDS forum here for more info.