The best way to learn the importance of using the right onion is by making french onion soup with red onions.

You'll quickly realize that it belongs straight in the trash.

Do you have a recipe to recommend?

I've very often made it with red onions (or half and half red vs white/yellow). It's delicious and always been a success.

I've used a version of this recipe [0] which explicitly says "red or yellow onions". It's also honest on the cooking time. I'd agree with its main warning, that making your own nice stock from bones really improves the flavour. Doesn't have to be beef stock. But I've also used stock cubes in the past.

[0] https://www.simplyrecipes.com/recipes/french_onion_soup

    I've never really figured out the difference between cooked yellow, white, and sweet onions

I agree for high-heat stir fried dishes.

tl;dr When cooking, use yellow. They're cheaper and you need less.

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Yellow onions have a stronger flavor, even after cooking. (Though cooking of course lessen it a lot.)

White and sweet are essentially the same, with the latter being slightly sweeter.

In the end they do taste similar cooked. Biggest difference is price. Yellow is usually 20% cheaper than white and 40% cheaper than sweet.

Not even close - sweet onions (like Texas 1015s, Vidalias, or Mauis) may be whitish, but they have noticeably more sugar embedded into the structure/tissues of the onion itself. Sweet onions are not white (or yellow) onions, or vice versa. FWIW, Vidalias are a bit less crisp than the others, which helps if you're really after carmelizing, but I find 1015s are the "sweet spot", so to speak, of softenability and enough fiber to hold up so they don't turn to mush.

Red onions are a different flavor profile altogether, and best for some (not all!) Mediterranean or Mexican dishes, or smoked salmon (NOT lox - yuck!) on a bagel with cream cheese and capers.

they taste differently from each other. some are sweeter and some are less sweet, due to having less sugar. Some are sharper or more bitter than others.