The most important thing about posture is that you actively monitor it throughout the day. Meaning you have to learn to cue yourself to sit up straight, shoulders back, chest up. No amount of working out is going to fix years of daily bad habits. Just keep reminding yourself to maintain good posture no matter how annoying or trivial it seems at the time.
Sitting at the computer all day tends to "pull" the head forward over time, and the chest muscles get shorter while the back muscles elongate a bit--at least this is my understanding. This produces the hunched-over look we commonly see today.
That said, I used to have really bad posture (still dealing with a little bit of forward head tilt) and the thing that helped my posture the most was probably lots of back work. I did (and still do) many, many pull ups, lat pull dows, barbell rows, and face pulls. I also de-emphasized chest work for a while... I did about twice the amount of back work vs chest work at the time.
Honestly, I thought your pics would be a lot worse though. I'd say you're good to jump right into a basic workout routine, but I'm definately no expert. I think worst it might do for you is nothing.