Okay due to the very confusing, contradicting nature of the information on the internet I will attempt to explain how to Softmod your Wii by installing the homebrew channel.
YOU CAN BRICK YOUR WII IF YOU MAKE A MISTAKE. BE EXTREMELY CAREFUL.
There are several ways to go about this. Prior to the 4.0 Wii System Update, I preferred the Twilight Princess Hacked Savegame as it was pretty easy to do. As of 4.x it no longer works, though. If you have system menu 4.2 or earlier, you can use Bannerbomb, otherwise you will need a copy of Lego Indiana Jones or Super Smash Bros. Brawl.
Since the current Wii system menu is 4.3, and Bannerbomb doesn't work on that, I'll focus on these:
The Lego Indiana Jones hack is known as "Indiana Pwns" because, hey, this is the internet. You can read more about that here, on the off chance you own Lego Indana Jones but not Brawl.
The Smash Bros one is known as "Smash Stack" because it's a stack overflow error and it's a programming joke only nerds will understand. Also if you're not a nerd you were probably going to be really frustrated by Twilight Princess and Bannerbomb anyway.
I'm focusing on the Smash Bros Brawl one, because Brawl sold alot more copies than Lego Indiana Jones (Although Lego Indiana Jones is a good game too) and was the reason alot of people bought a Wii. It also skips the Wii's internal hard drive entirely, so there's not to much screwing around with the Wii Settings-- Smash Stack essentially loads a custom stage that's actually the homebrew channel incongnito, and that causes an overflow allowing the code to run. This also means it'll be hard for Nintendo to patch.
You're gonna need an SD Memory Card, NOT SDHC. This means less than 2 Gigs in size. Smallest I've done this with is 256 MB. This memory card will be in your Wii most of the time, so it might be good to buy a new one for this, otherwise you'll need to switch it between your Wii and digital camera constantly and that's annoying to keep up with. Be sure to format it in FAT32. Any other Filesystem does us no good here.
You're also gonna need something that can open Zip Files but since this is no longer 1998, and every version of Windows since XP can do this natively now, I'm not gonna stress it. I prefer 7-Zip because it supports every file format ever invented, but you can use WinRAR or WinZip if you want.
Okay so what you have to do is move ALL your custom stages from your Wii System Memory to the SD Card. This won't work if there are ANY on the Wii's hard drive. Also be sure to shut off "Smash Service" if you have it enabled and delete those stages. If you have any of them you have to wait 24 hours after deleting them and shutting it off in order to continue.
Back up any custom stages to your computer for now, and format the memory card as FAT32.
Download the smash stack zip, the link is here below the Smash Stack Logo. Next, copy the private folder from it to the root of the SD card.
Download the HackMii installer from bootmii.org and put it (boot.elf) on the root of the SD card.
Put the SD card back in the Wii, start up Brawl, and go to the Stage Builder.
At this point boot.elf should load and HackMii will start running.
After a minute or two, it'll say "Press 1 to continue" at the bottom. At this point, it'll load a screen where you can install The Homebrew Channel, DVDX, and BootMii.
DVDX and BootMii are optional, but definitely install The Homebrew Channel. It's your new best friend.
DVDX is DVD Playback for the Wii which is something that Nintendo really dropped the ball on. Since I watch movies on my computer, I've never used it, though.
BootMii, however, is pretty useful. In case you destroy your Wii System Menu, BootMii can restore it if you install it. BootMii allows you to run homebrew before the System Menu even loads. If you have BootMii installed there's a pretty good chance you can unbrick your Wii if you do somehow Brick it. I've heard it's not compatible with some of the more recently made Wii's though.
But what we want right now is the Homebrew Channel. Go ahead and install it, it's the single most useful thing your Wii will have.
After that, you're good to go. You can go here and find any application you may need, although that list is kind of incomplete. You can find alot more Homebrew Programs on GBA Temp's forums, but the problem is, the information is surrounded by a sea of one time posters asking stupid questions.
Anyway, once you download an app, just put it in the apps folder on the SD card. It should be like, F:\apps, or something similar.
I'm gonna go out on some kind of crazy limb here and say you want this. This is NeoGamma. It lets you play "backups" of Wii games. Luckily, it has very few issues. I've been using it to play Metroid: Other M which is a fairly recent release.
I'm not answering questions about your specific Wii. I'm not Jesus Christ, I've come to accept that now.