I am also new to the world of DS consoles but recently bought 2x DSi for my kids.
From what i know the main differences and there may be more:
DS Lite - no camera
DS - has camera
DS LL or Max - has camera and bigger screen
The R4 card you refer to is a card you put into the game slot and you can load a micro sd card into it and load onto it "backed up games". I purchased an 8 gig card and with games averaging around 100mb or less you can fit a lot on them.
Im sure someone more knowledgeable than me will post more info.
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