Piano is a great instrument pick up – I learned the basics when I was really young from a pretty brilliant teacher, which was very lucky… that was the basis for all my musical learning… it taught me to sight read and whatnot.
As far as playing “by ear” this is totally something you can learn, but it requires a few different skills. Like Steve said – you need to be able to hear the intervals. You also need to be able to pick out the key of a song – not all the notes, right away, but the key is important. Once you know this, and you know the notes in each key, then you’re on your way because you can pick out the first note, hear the interval – and even if you can’t quite tell if it’s a half step or a whole step, you should know if it’s the next note in the scale, or two away, etc. By-ear playing takes a lot of practice, but it’s certainly a lot of fun. If you’re looking to do more with it than just fun, then I’d say take a few lessons, even if just so you can read and write musical notation. Finding a private teacher that knows how to play by ear might be useful as I bet they have some exercises to aid in learning that.