Going Beyond Beginner Piano Level
You can be taught how to play beginner piano in a relatively short time. After about two or three lessons with a professional piano teacher you ought to have learned where every one of the notes are and as well how to play certain beginner tunes, perhaps even with 2 hands together. I myself have put together a video teaching you how to play the basics of piano in less than 10 minutes. Take a look at this short video here – how to play the piano for beginners.
Moving on and playing with both hands as one is when the whole process begins to get a little more tricky. For those who are being taught with a classical piano teacher they will no doubt be given a one or two of set pieces to learn. You will learn these pieces by first playing the right hand part on its own. Playing it time and again until you can play it without indecision. After that you will do exactly the same with the part for the left hand. For a lot of people this is a little more difficult to achieve however following plenty of repetition it ought to become easier.
Then comes the difficult part… Playing  both hands at once. This can be quite like learning to ride a cycle. Sooner or later it will all just come together but until then, you will find the learning process a little slow.
For more information on piano playing and for a list of some of the best home study books to be had take a look at the Learn Piano website.
You will be learning to play in several styles or genres at once if you have chosen to use one of the home study courses available. The basics of piano learning remain the same though. You will need to learn a simple tune or some simple chords in the right hand and a simple bass line in the left. Getting the rhythm of the bass line and the rhythm of the chords (which will be different) to merge will not be as difficult as combining the hands whilst learning set pieces. You will be playing quite a lot earlier this way. You could also look at the following site for extra information on learning piano at home: How to learn piano
On the subject of advancing in these styles you will have to include a tune in the right hand as you play the chords as well with the same hand which will be very tricky to begin with. Or instead sometimes you will be playing just the melody in the right hand and you will be combining the bass line and the chords in the left. Once again, the combining of two things with one hand is what you will find more difficult but in the end this technique is more favorable to speedy advancement than going to a private teacher.
You should certainly consider using a professional piano teacher as soon as you have worked through your home study course and have attained an intermediate level of playing.