I am, sadly, notorious amongst my friends, for my wine spills.
I tend to talk with my hands. And I tend to have a glass of wine in one of them. And that glass is usually red. This obviously, is a situation leading to the occasional minor stain.
In general I spill copious amounts of soda water on the stain and blot. After a while the shadows of these stains tend to accumulate, and at such points I rent a Rug Doctor or equivalent and clean the apartment.
I have tried other methods (baking soda etc.) but none worked as well as the above. Today I found something better than all of the above: Bissell’s Little Green Machine. I bought the turbo model with heating, but I doubt you need it for red wine. Vacuum up the wine, rinse carpet and vacuum again. Maybe 10 minutes work for the average spill and the carpet dries like new.
Bissell makes a hand vac version, the SpotLifter that may be all you need. I sprang for the slightly more expensive machine, but then I spill at a professional level.