white wine or white vinegar
baking soda (optional)
First, spill the red wine on the carpet :)
Once you've got the spill, remove as much as you can by dabbing with the towels (do not rub, though!). This dabbing takes a while, so be patient.
Once you've gotten as much out as you can, spill white wine or white vinegar over the spot, let sit a few moments, then begin dabbing again (again, press the towel down on the stain, but do not rub).
Repeat this step for as long as your towels are still coming up pink and you can still see a stain on the carpet. This works really well, but you just have to be patient.
When it's 99% gone, you can pour baking soda over the spot and let it dry, and vacuum it up 12-24 hours later. This will often remove the last lingering shadow of a stain.
Actually, if you spill red wine and are out of white wine or vinegar, you can just pour baking soda on the spot, let the baking soda absorb the wine and dry (which can sometimes take a full day or two), and then vacuum it up when it is dry. This method works pretty well, too.