Firstly, bad sitting will lead to bad posture, which will also be evidenced by a weak core.
Secondly, no exercises will let you sit properly; all they can do is to strengthen your core and give you good reasons to sit properly. You have to make a conscious effort to sit properly. After practising it for a while, it might become second nature to you; however, it's still up to you to continue.
Thirdly, when you initially start sitting straight, your back will become sore very quickly. This is normal; the soreness will gradually wear off the more you keep sitting properly.
Tips For Sitting Properly
Endeavor to always stand up straight. As ironic as it sounds, it works. If you observe soldiers, the way they sit and stand are identical; they have to sit straight. This is because it allows them to quickly react. The more you stand straight, the easier it'll become for you to sit straight. When standing, try not to lean on an external support, lean on one leg, or bend at the waist.
Remember to exercise your core: There are varieties of core-strengthening exercises and a lot of them can be performed without any equipment. Plank is a favorite; push up is another awesome exercise. Burpees, running, jumping jack, and legs up are great ones too. Even high-tempo dances are great for the core.
If you can, alternating between your regular chair and a stability ball such as
is an awesome way to strengthen your core. Not only is it playful to sit on, it actually forces you to sit straight (most of the time). Don't be surprised if your back becomes sore after sitting on it for less than an hour.
Sit straight in your chair! Unless you're sitting on a sofa or a very-cushioned chair, most chairs have a rigid structure that allows you sit straight. Make sure that your butts are touching the back of the chair and simply sit straight. When your butts touch the back of the chair (or is as far back as naturally possible), the only ways for you to sit improperly would be tilting the chair backwards or bending your chest forward. You can easily correct those once you realize you're doing them.
Tidbits Not To Forget
Sitting properly is very easy to achieve as long as it's a goal.
It'll initially be hard, but it'll get easier as you continue with it.
Don't go gungho on it. Do it gradually. If you use the ball, you can alternate (between it and your regular chair) every couple of hours. You're also allowed to cheat; I know that some people put pillows or soft covers on the back of the chair so that although they're sitting straight, their backs are actually resting on the pillows.
Have fun with it. You can even perform some quick core-strengthening exercises on the ball. Enjoy the process and make it fun. :)