I have the same problem. I read consistently that a lot of poor posture has to do with weak core muscles, as Miss Inconnue suggests. I intend to fix the problem by getting professional help, that is, a few private sessions with a personal trainer or of pilates.
Knowledgeable men, however, tell me you can also find good exercises on YouTube, or your friends at the gym might have suggestions.
Meanwhile, when standing, do NOT hold your hands with arms mostly straight, clasped in front of you. This slouches your shoulders forward, making you look meek and shorter. It's called the "fig leaf pose." At least hold your hands behind you, though that can make you look like a guilty schoolboy, but a tall one.
+1 on the suggestion of deadlifts, lat pull downs and I would not forget ab crunches on stablity ball, you want to look after the whole core. Just make sure with deadlifts your back is straight, not curved or you will be on a first name basis with your chiropractor. Start with light weights to get the technique right.
Yoga is a good addition to the workout regime and changes things up. Speaking from experience from my first class this past week, it's not as easy as it looks and you will be stiff later in the day.
No. I spent four years in college either hunched over a desk or a poker table. I still catch myself slouching on occasion. Start making it a point to lift heavy things. Build up your upper body strength. Any cardio should help with the breathing. Mainly, when in public, remember to suck it in, and keep your shoulders set. After a few months of consciously working on it, it does start to become a habit. After a while, you start feeling like a schmuck if you are slouching. It's really all about the will power to keep at it.
One other thing I've noticed, cowboy boots seem to make good posture seem more natural. Something about the higher heel. If that's your style, it's a good cheat.
+1 to a heel
I'm going to lose my girl-card for revealing this, but wearing high heels usually makes for a physically happier day because it corrects my posture. Of course, my heels are 3" max, and my favorite pair are the best-made shoes I own, so it all depends.