I'm going to assume you want to know about digital photography methods in general, rather than film-and-darkroom techniques.
When I look at that picture, I notice two things. The colors are very saturated. The model's skin tones are a bit green. I don't know what the original photo looked like, so I can't be sure what changes were made. I'm just making a couple of educated guesses.
One way or another, more green was added. There's a lot of green in the picture overall, so cranking up the saturation would add lots of green. On the other hand, in an RGB image, one of the basic color channels is green.
I would guess that either the green channel was boosted, the overall saturation was boosted, or both.
The reason that this looks "old-timey" to you is that, when film photographs age, some of the colors fade faster than others. That makes the slower-fading colors stand out more. Boosted green makes it look a bit like a film photo with reds fading faster.
If saturation is turned down, colors move toward grey. Turn saturation all the way down, and the result is a black-and-white picture. If saturation is turned up, colors move toward rainbow-like purity. Turn saturation all the way up, and the result is cartoon-like, looking too pure to be real.
If you have photo-manipulation software (like PhotoShop, but there are tons of 'em, some of 'em free to download) you could look for how to change "gamma" on individual color channels ('cause that's how you can boost just the green) and you could look for how to change saturation ('cause that's how you slide the whole pic toward cartoony-ness). Playing with those two things might get you the effect you're looking for.