There are only a tiny handful of 'DSi games', games which specifically need the DSi's camera to play. Most are even DSiware. So technically all DS game cards will work just fine on a DSi, there's no need to do anything to get them working.
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Yes, you can play DS lite games on the DSi as well as the DSi XL
You don't need any additional software, just purchase and download games from the DSi shop.
I have just been lookin on the Nintendo website and you can play your DSiware games straight from the SD Card + you can copy DSiware game save files to your SD card. You can't however transfer your DS games to an SD card or transfer your DS game save files to your SD.
Normally, you can play new DS games on old DS. But if the games says for DSi only, or for 3DS(will be pubulished soon) only,the it can not play on old DS.
For the most part, yes you can. If you want to connect with pictochat or DS games (including DS Download), you can connect between a Nintendo DS and DSi. However, for DSi specific games, such as DSi games and certain applications, the Nintendo DS will not be able to connect. But, note that certain DSiWare games that can DS download play still can be connected to the Nintendo DS.