Lots of info, thank you. I know very little about this stuff as is, I mostly just plug my Wii with component cables into my TV for Gamecube games or use my shitty av to HDMI converter for PS2. Both look VERY dark and dull. Not sure what option works best for that.
So a lot of TVs don't do great jobs at upscaling 240p sources (which will be a few PS2 games, not many GC games). They should do ok jobs with 480i and 480p though. When you say AV to HDMI converter I assume you're using composite with your PS2? That's probably a big part of it. If you don't already have them I would highly recommend grabbing some component cables cables for your PS2. HDRetrovision makes some, and you can also use PS3 component cables on the PS2 (stay away from 3rd party ones most of them are made pretty crappily, with the exception being HDRetrovision).
In your case I would recommend trying something like an OSSC or Framemeister. Like I said the GC/Wii solutions right now are just 480p so you'd be in the same position you're in now, which is relying on your tv to upscale from 480i/p. That's why I would also say to skip something like the retrotink or rad2x. Since both of those are again only going to 480p (which you can get out of the Wii).
The framemeister will give you more options IMHO, and will work with more tvs, but it's also more expensive. It's currently around 340 bucks, and it's also end of life because the main scaling chip in it isn't being made any more, so they're making what they can with the supply they have left. I also think the framemeister handles 480i much better than the OSSC, but I personally can't stand bob deinterlacing, the up and down motion stands out to me like a giant vibrating sore thumb, but to others its a lot less noticeable. The framemeister will also let you plug in, composite, and s-video along with RGB and component. Since the framemeister is an upscaler, it has a framebuffer, so its able to output everything at 60hz so you don't run into weird refreshes or incompatibility issues. It also means you can do somethings with zooming and stretching the image that a simple line doubler can't.
The OSSC is cheaper and going to run you like 150 - 200 depending on where you order it from, and if you get it with a remote and power supply, etc. The OSSC is a line double and not an upscaler (its accomplishing a similar thing but in a different way) so you have 0 lag with the OSSC (framemeister ranges about 1 frame to 1 1/2 frames depending on what you're doing). Though compatibility between different tvs and the OSSC can be spotty. Since its just line doubling you end up with some weird resolutions and sometimes off spec frame rates that some modern digital tvs just don't like. So you can run into an instance of your tv only taking line 2x mode for a 240p source and you're back to just 480p. Plus the OSSC only does Component and RGB in. As I said before I am not a fan of the OSSC's handling of 480i sources but it does handle 480p (which it line doubles to 960p) better than the framemeister upscales it. I do believe you can google ossc compatibility and find a google doc with people saying what tv they have and if it works with the different 2x,3x,4x, and 5x modes of the OSSC.
Though do realize that in a modern setup the different modes need to compatible with everything in the chain. So if you have an AV receiver that everything runs into, it also needs to be able to handle the signals coming out of the OSSC.
I hope this helps you a bit.