anon is right about salsa just being another dance in which you practice and become proficient in. You just have to find an instructor who can teach you, and be willing to devote yourself to it. As for swing, it's a really fun dance, but I have to admit, the swing clubs are usually filled with the over 45 crowd, so for a young guy trying to use dance to pick up women, swing might not be the best thing.
With that said, I feel it's important to point out that the vast majority of women in dance studios are not there to meet guys, and I'm talking about chains and independent studios. If they wanted a guy to pick them up, they wouldn't hang out in an environment where men are scarce. I've seen guys try to pick women up in a studio, and it's just pathetic. They're not that focused on the dancing, and it shows. I think to use dancing as a means of attracting women, you've got to be an amazingly good dancer. The guy who does a few groups and takes private lessons once every couple weeks, isn't going to develop the skills to be really 'natural' as you suggest. I think Hollywood has created this notion that a guy with dance skills is automatically guaranteed to get women. This is such a farse. Being able to dance just adds another dimension to yourself. That may make a guy more attractive to women, or it may not. I once brought a date to a dance party, and she just got angry with me because I could dance and she couldn't. The fact that it takes years of training to achieve some ability will discourage most guys who just see dancing as a pick-up tool. Think about it, if taking lessons for a short while really was this secret way of attracting women, dance studios would have three times as many men than women.
Dance because you love it, and you get something out of it. Get serious, and work hard to improve. Don't eliminate learning a dance just because you think you won't be able to use it as a dating tool. Who knows, you might meet a girl who's mother loves swing. Wouldn't it be cool to impress her too? Someone who dances primarily for any reason other than the fact that they love it, will never appear natural or be a good dancer (competively or socically).