Currently you can't softmod a DSi.
To play all those games you need what is called a flashcart. Good flashcarts are:
Acekard 2i (AK2i with AKAIO installed)
EZ-Flash Vi
M3 Zero
Supercard DSONEi
Supercard DSTWO
iEdge
Those that suck or have little to none support:
Any R4 whatever clone
DSTTi
N clone with the i in it.
The respective subsections for those types of cards are here:
http://gbatemp.net/index.php?showforum=37
You can ask what to get here:
http://gbatemp.net/t217593-the-which-flash...get-info-thread
you might get a fanboy response, ask at your own risk. You should also ask where to get one, if you want a local source or get it cheap from a shop in Hong Kong.
If you want to jump head first into things, just get a AK2i. It is cheap so if you think it is not for you, you don't break the bank getting one.
Then you will need to get a microSDHC, maximum capacity is 16GB unless you want to spend 200 bucks for a crappy class 2 32GB microSDHC from sandisk. You should get a class 4 or class 6.
As to how to format that, you shouldn't need to reformat a microSDHC when you get it. In rare cases it will screw up the flashcart if you do. NEVER use the windows formater to format your microSDHC. Default format is FAT32.
Ask your question in the which flashcart post then visit the subsection for any customization:
http://gbatemp.net/f40-nds-flashing-hardware-and-software
You asked in the wrong section, a mod will probably move it soon.
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Ignore post since it is a wii and not a dsi.
You will need to softmod your wii, if it is not already done so. Once that is done, you will need to install the usb loader. Once that is done you can prep a HDD to connect to the wii and use.
Links you will need:
Softmodding your wii:
http://gbatemp.net/t171901-modify-any-wii-4-2-below
Backup loaders basics:
http://gbatemp.net/t179085-9658-9658-9658-...-9668-9668-9668