I find that many pianists who are new to the instrument, and even some more advanced players, have what I call a white-key mentality. To them, playing on the white keys is "normal" and playing in keys with sharps or flats is strange, even scary. Beginners at the piano spend far too long playing in just the key of C and develop a fear, or at least an unease, about playing in other keys. This is a detriment to mastery at the piano.
If you stay playing in C for too long, you may also develop the habit of trying to play at the edge of the keys. When you play in keys with sharps or flats you must be farther in (towards the fallboard) for your hands to be comfortable. But you must develop this habit early.
When I have a new student come to me for lessons, whether brand new to piano or an adult who played as a child, I always ask them "Why do you think we have black keys and white keys?" The absolute beginners have a good excuse not to know, but I find even players with some experience don't really know. In fact, most people have never even thought about it, nor did any previous teacher explain it. Many people will give answers such as: "they are the sharps and flats" (yes, that's what we call them but it doesn't explain why we have them), or "they are the half-steps in between the white keys" (there are white keys which are half-steps as well), or "they are the minor keys" (totally wrong, because the sound that we often associate with the word minor has to do with relationships between tones, which can be between black keys or white keys, and it not related solely to the black keys at all), or "they are dissonant" (again totally wrong because dissonance, two or more tones that clash with each other, is a function of the mathematical relationship of tones' frequencies and can occur between white or black keys). Many students will swear that the black keys, as a group, sound different than the white keys; however if they close their eyes I test them they cannot tell black keys from white. I then show them the inside of the piano, where every note is a string (or strings) struck by a hammer, and looking at the strings you could not possibly tell which belongs to a white key and which to a black. It's strange that people would hold these misconceptions, if you just think about your voice, or another instrument such as a violin or clarinet or guitar. There are no white keys or black keys on those instruments yet they play essentially the same types of music as the piano (only the melody in the case of voice, violin, etc.) People are usually quite surprised to learn that the black and white key arrangement of pianos and all keyboard instruments is purely to serve as a visual aid. Just cover up the black keys and you'll see the white keys all look alike, and you would have no idea which was which. The very ingenious arrangement of the black keys in groups of two and three enables all 12 distinct keys within the octave to each have a unique look by virtue of the other black and white keys around it. Not to mention that if there were no black keys, the keyboard would have to be quite a bit longer (or the keys narrower) to fit 88 keys within a span the arms could reach!
When beginners start out I do have them play on the white keys, in the key of C. They play simple songs by ear (see my post on the importance of playing by ear). But then as soon as possible I have them learn other scales, starting with G and F, with a very easy fingering (splitting the scale between hands), so they get familiar with these, and then play their songs in these two keys as well. That means also learning the I, IV, and V chords in each of those keys. Then we move around the Circle of Fifths until we eventually get to all keys. I cannot stress the value of this enough. Not only does it get the hands familiar with the different feel of various keys, but you are using your ear as well. When you have already played by ear in other keys, your hands just know the feel of them and it won't be difficult when you start reading in those keys.
Another tool that I stress is improvising in all keys. You must know each key so well that you just feel you are immersed in that key, that you "live" in that key. (It's just like a foreign language that you know so well you begin in "think" in that language.) Later, when you read a piece in that key it will be so natural you won't find it any effort to remember its flats or sharps. One thing that some students do is to circle all the notes that are flats or sharps. This is a terrible idea. It creates dependency on the visual cue instead of using your auditory and physical skills. And what will you do when you play in the key of F-sharp with 6 sharps, circle almost every note?
The different keys (tonalities) each have their own character (otherwise why wouldn't composers writing for piano have written everything in C?) Try to develop mastery in all keys so you don't have a white-key mentality.