MANY PEOPLE ARE MESSING UP BY NOT FOLLOWING EVERY STEP!
FOLLOW ALL OF THEM EXACTLY AS THEY ARE!
Note that the best firmware for Homebrew is 3.2
First of all, you will need to install the Homebrew Channel (HBC)
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If you need help installing the homebrew channel, refer to Nito711's vid response Or use the twilight hack.
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DOWNGRADING WITH 3.3v3 OR 3.4
I added some new links.
After the HBC is installed on your Wii, delete everything on the card.
Step 1 - Download these files:
http://www.megaupload.com/?d=8OF7W3BP
Step 2 - Extract the wii downgrade.rar and drag the "apps" and "wad" folder on the SD card.
Step 3 - Using the Homebrew Channel, open WAD Manager 1.3
Step 4 - Press A on the Wii Mote once. The Wii should detect a SD Card. Press A again to see a list of WADs on the SD Card. Use the D-Pad to select IOS16-64-v257-prod.wad. Press the + to install the WAD. Once it has installed the WAD, press the Home button to restart you Wii.
Step 5 - Using the Homebrew Channel, open Custom WM. Now using the same method as in Step 4, install the cios_fix.wad. Once it has installed the WAD, press the Home button to restart you Wii.
Note: If that fails to install, just use 'Wad Manager 1.3'
Step 6 - Using the Homebrew Channel, open IOS Downgrader. Just follow the instructions and wait until it finishes installing the necessary IOSs.
NOTE: You're Wii does require connection to the Internet otherwise it won't work. Same for the next step.
If you do have a Internet connection and you get errors while it's installing the IOSs, just restart the Wii and try again.
Step 7 - Using the Homebrew Channel, open Install Shop IOS51. Say Yes to installing the IOS51 and Yes to patching a fix but No to the Wii Shop Channel (it shouldnt matter if you install but it wont make any difference).
Step 8 - Using the Homebrew Channel, open WAD Manager 3.
Step 9 - Now you should be able to install WADs. Try installing the WAD MANAGER V3.wad
Step 10 - Downgrade to 3.2 or whatever firmware you want. Make sure you choose the right region or you will semi-brick your Wii. (U = USA, E = Europe, J = Japan)
Use this file with the HBC to downgrade. Make sure you put it in a folder in the 'apps' folder.
You can also use AyRegion Changer to downgrade.
Your Wii is now downgraded!