You should start by learning to play scales and arpeggios (as mentioned before).
Once you can play these reasonably smoothly, I would strongly advise you to start playing Bach's Inventions (at a relatively slow speed at first). These are real technique builders for both hands, while allowing you to develop your dynamics and musicality.
You should NOT play "exercises" like Hanon exercises.
It will give you repetitive strain injuries in no-time at all, since most of these exercises
focus on playing the same exercise time and time again, using the same muscles and tendons. (trust me on this, I have seen people go from playing pain-free to not being able to lift a cup of tea because of these dreaded exercises) If you don't have a teacher, playing these exercises is especially bad, since you could be ingraining the wrong technique. Because of the numerous repititions, it'll take a huge amount of time to get rid of the wrong habits. Another reason why you should play songs instead of exercises is that the exercises are incredibly boring and you don't develop dynamics or any sense of musicality in your playing. Excuse me for the ranting, but I have seen so much trouble caused by these "exercises" that I don't want you to make that mistake.
If you can get access to a teacher, I would strongly advise taking lessons. Playing the piano properly requires teaching, your posture and technique are very important. If you play piano using the wrong technique, you can develop injury very quickly in comparison to other instruments. Also, if you have access to a real acoustic piano, I would advise playing that, since it's better for your technique.