I'd like to dump here several of my findings while to get the best picture out of the Wii without any fancy scaler or stuff like that.
The quality build of Wii2HDMI adapters is very varied, and it's pretty much a roulette. You can get a decent one that simply transforms the 480P analog signal into digital, and those offer a similar picture quality of your average Component cable. But you can also find all types of horror stories about interferences, bad colors, no sound, some stretch the image to 16:9, even if you already had it at 16:9 and such, so you should watch out before you buy.
The decent ones I tried gave a slightly brighter picture than the component cables I have (which is nothing special, I'd like to know if there's much variety in quality with component cables) but besides that they're quite comparable. The major disadvantage of HDMI adapters is that they're not compatible with all the resolutions the Wii can handle, specifically the resolutions under 480i, like 240P which can be used by Virtual Console content and emulators. Another weakpoint is that it takes more time to get the signal back whenever there's a resolution change. But if your TV can only be plugged with HDMI cables they will have to do.
Of course, there's some HDMI adapters that scale up the Wii's picture (not the system's resolution!), I never had any experience with those so I can't comment about them.
The advantage of Component cables, if you can use them, is that they'll work with any resolution that the Wii can use and the change from one resolution to other is much quicker. Although 240P is also dependent on your display, not every TV (specially modern TV) is capable of 240P, and even then only part of what the Wii can offer is 240P, the rest is (or should be) 480P.
Also, I don't know if it is widely known, but the Wii looks better in 4:3 than in 16:9. It's true that having a larger field of vision may seem comfortable at first, but if you compare image quality, 4:3 is noticeably less blurry, and it's very easily to spot with text or any sort of 2D texture, like HUDs, menus and such, At least to me "more screen" really doesn't pay off if it's going to look worse.
In fact, setting the system to 4:3 is what yielded the biggest gain in quality for me of all the things I tried.
With all that said, I kinda wish I had a good CRT with component input, or there was an "ultimate" monitor for old systems such as this one. If there is, I don't know it at least.
EDIT: Hi, I'd like to add a bit of information about the 4:3 vs 16:9.
First of all I'd like to point out that not every game was designed for the Wii to work at 4:3, a few of them are intended for 16:9, those being mostly 2D Platformers with some focus on multiplayer, such as Donkey Kong Country Returns, New Super Mario Bros Wii, and Kirby's Adventure Wii/Return to Dreamland.
There's also at least one that changes from 4:3 to 16:9 depending on what is displaying. The Kirby 20th Anniversary compilation initially is designed for 16:9, as it contains gameplay challenges based on Kirby's Adventure Wii and the museum, and 4:3 content like the actual Virtual Console games.
These games in 4:3 will be letterboxed, and everything will look tinier than intended, so those are better in 16:9 even if text and such is a bit blurrier.
But for the most part, I have to say, what a change going from 16:9 to 4:3. Games like Smash Bros. Brawl look so much better, Wario Land: Shake It is beautiful today (although 16:9 only adds some borders to the side), The Legend of Zelda: Twilight Princess is also much sharper, although it is to be expected, as I found out that the widescreen used by the Wii and other non-HD systems is actually anamorphic, basically stretching the image to fit 16:9.
This finding is Kind off a mix between disappointing and great, as I liked the bigger screen, yet games looking better is a big plus. I'll always choose better picture quality over size (just like I'd always choose frame rate over graphical power, lol).
The only time I'm somewhat reluctant to use 4:3 is Twilight Princess if only because the added field of view (and moving away the map and HUD elements) to have a clearer view is pretty useful, but oh well, no biggie.
