At OP: First of all... clean that ashtray you call a Wii. I'm assuming it's equally horrible in the insides. So, take it apart and clean it, if there's dust on Capcacitors, they might overheat, reducing the lifespan of your ashtray.
Once you've cleaned properly everything inside, it's time for the software side of things.
First, "rehack" your Wii by using any method of your choice. By "rehacking" you'll update the Homebrew Channel, which is very important. After you've reinstalled the Homebrew Channel, make sure to install PriiLoader (avoid BETA versions) and BootMii.
Make a NAND copy with BootMii, and make sure you can access PriiLoader by holding RESET while turning on your Wii.
Then, install cIOS d2X. Some people say you no longer need to install Waninkoko's and Hermes cIOS, but I personally never had a problem with my Wii and I strongly believe it's thanks to those two cIOS (Waninkoko + Hermes + d2X) so I can't recommend this enought, but I do suggest people to install Waninkoko and Hermes too alongside d2x... of course, if you want to.
After these steps, you are pretty much free to do whatever you need with your console... in other words, install games, channels and what not.
Be aware that some homebrew apps needs cIOS versions for them to work properly, one example I can think of is Riivolution, that needs a special cIOS installed in order to function properly, another example I can think of is IOS60 that enables loading channels from SD Channel in Wii Home Menu. On this note, emuNANDs seem to be a popular choice over official SD Channel, so that's another option (emuNAND).
Regarding old console emulation... there's been many achievements in the last years, for instance, almost full compatibility with PS1/PSX games. So many branches/forks of the old branches/forks of the original emulator that It's pretty honest quite confusing to keep up with the latest versions. But I utilize WiiSXRX Preview 4.0 version and I haven't had any problems with PS1 games.
RetroArch is interesting too, since it allows you to load old Arcade games now from the Full Non-Merged MAME2003 romset, and also I'd dare to say 99% compatibility with NEO-GEO roms.
CPS3 emulation is possible too, but it's choppy.
Regarding other systems, such as CPS2, CPS1, FBALPHA and MAME2000, it really depends on the version of RetroArch you download. Here's
my old RetroArch Wii "guide"/"tutorial" if you ever want to try Arcade games.
Regarding other systems, such as NES, SNES, GB/GBC/GBA and SEGA Genesis, you're better off with the standalone emulators rather than using RetroArch's cores for said systems. It's a hit or miss thing with speed/audio issues/random crashes... but then again, my RetroArch thread will give you an insight of how RetroArch Wii behaves. You don't have to read the whole thread, just use the Threadmarks as I made important posts regarding RetroArch updates.
For N64, you're better off using official Virtual Console games (21 N64 titles) and you can expand a little bit your N64 library by utilizing Not64 and Wii64. Both has different compatibility issues, so use both if you want.
Nintendont is the GameCube loader by default. It's pretty good, never had an issue with it. So I can't recommend that enough times. It even allows you to load TRIFORCE Arcade games!
Regarding Wii games... I use both USB LoaderGX and CFG USBLoader. Never had an issue with both, and if I'm honest, I prefer CFG over GX, but lately I've been using GX a lot more than CFG because of some video improvements.
Other things you can try are ports of old games to the Wii, which won't be hard to find if you use WiiBrew or take a proper extensive look at the Wii Forum. You might get surprised at all the things the Wii is capable of doing nowaadys.
EDIT: Grammar issues.