Soup – make French onion soup with red onions instead of white

Typically you should use yellow onions for cooking. They have a higher sulphur content and are more flavorful after cooking. Raw, a red onion will taste more pungent. However, once cooked it is more mild and sweet than a soup would require. Red, as well as other sweeter onions, have more sugar and water than their yellow counterparts. For more details on the profile, seriouseats has good primur.

If you wish to improve your results, you might slice more thinly and add some salt in advance of sweating (rather than before adding broth) to draw out the excess moisture, as the onion's moisture (more prevalent with red onions) will inhibit browning, ergo inhibiting caramelization and the maillard reaction that break the bigger sugars into little ones and lend flavor through browning. The explanation at norecipes also adds that deglazing with sherry can slow down the process if you are experiencing uneven browning. Another possible tactic would be adding a 1/4 teaspoon of baking soda per pound chopped onion; this will speed up browning.

Ultimately, unless the recipe calls for red onions, you are going to get better results with yellow onions in terms of raw, "sweet and tangy" onion-y flavor. You can use the tips above to try to overcome that, but a nice big bag of yellow onions are going to do you better in the long run, and if you put the same effort into the it will take less time, taste better and be less expensive.

I would simply use whatever onions I had available! Red onions have a milder flavour than yellow onions (therefore red is usually preferred in salads) and obviously different colour but the difference is still small enough to interchange with no problem. Unless, of course, you desire a particular colour in the soup or are particular about the soup having a strong or mild flavour. Leek would make it even milder and sweeter. If I'd have to make a choice I would use yellow onions and spare the red ones for raw/decorative use.