I have been self-teaching myself piano since mid-2006. One thing I've learnt from browsing the fora for beginners is that there is no one best way to learn piano. How best to learn probably isn't even static for the same person; what I mean is that optimal learning techniques must vary depending on previous experience, including what you've already learnt to play.
I started learning out of children's tuition books; I reckoned that they would have to explain things very clearly! Since I don't have a teacher to ask questions to, I end up doing a lot of research which I share below.
Everyone says that regular practice of scales is the best way to achieve this (NB: I almost never practise scales).
Otherwise, it's just practise, practise, practise!
This is just a curiousity issue. The default notation on each stave is for the middle-most keys of the piano, so how do you know when to play the very high notes with either hand, or outside the range of notes that fits nicely on the stave?
Actually, there are a few ways to do this
Sometimes so many notes are written to play on just one of the staves that you need two hands to play them; you'll see LH or RH written in at this point to tell you to use the other hand for the extra notes.
Weirdly enough once you master this it seems quiet simple. But I did find that I had to master it twice, once with the left as quiet hand and once with the right as the quiet hand!
At first, I found it easiest to listen hard to the notes (hand) I wanted to be louder, and to try to barely play the quiet hand at all. Even now, I concentrate on hearing the loud hand and almost ignoring the quiet hand -- it works well for me to achieve the desired effect. Some people talk about just resting the quiet hand on the keys and noting where they should play with it, or barely tapping the quiet hand while simultaneously exagerating the motions and loudness of the loud hand. Eventually, you manage to integrate simultaneous play.
A tie extends the length of the same note. You only play the note once with a tie, however long and however many of the same note are tied together.
A slur means running different notes together in sound, and it means playing a group of notes smoothly as part of a unit distinct from preceding or sucessor notes -- the main thing is that you do play all the notes as written (but look out for complications like tied notes in the midst of a slur). I'm a bit pants on getting a slurred group of notes to sound right.
What confuses me is why do ties and slurs have to look exactly alike?!
It takes me between 4 days and 5 weeks to learn new pieces. Supposedly, you get muscle memory going with instrument playing; your fingers/arms get to know what to do better than your brain does. And it's muscle memory that allows you to play competently. To speed muscle memory up, exaggerate the motions and play more loudly (I hate playing loudly).
Some piano gurus advise that one plays at full speed every time to learn quickly; other people find that they need to get the rhythm first, and they must tap out a whole piece several times. Once they have the rhythm the rest will come quickly, seemingly. People who play by ear (beyond me to even understand how that's possible!) have to listen to the piece repeatedly to pick it up quickest.
I find that I do best by playing the piece slowly but trying very hard to make sure that each note is correct; also, I divide the piece up into short sections (maybe just 2 or 4 notes) that I practise playing at proper speed, before tackling the next small group of notes (hopefully again at proper speed). Eventually, I can tie together different little sections to make it all sound right.
Also, I learn a new piece fastest and play it most smoothly if I try very hard to never look at the keyboard (see more on this topic below).
Moot points; blind pianists don't get to do it. They find their way around the keyboard by feeling the black keys and just feeling that their arms are in about the right place, too.
On the other hand, people who play by ear rely on watching their hands to see which notes produce which desired sounds.
I'm sure there's no right or wrong, you just do what helps you play best. I can't look at both hands at once, so I find it easier to try to not look at all and instead learn to go with "feel" -- I use one hand to guide the other, I fondle the black notes, I count keys to move along, and eventually I get used to moving my hand to roughly the right place or even exactly the right place without looking. I allow myself the indulgence of looking very occasionally when (typically a big jump on the keyboard is involved) I just know it's very hard to consistently get my hand to the correct place (in time, it's always a matter of time!) without looking.
The general consensus seems to be that anyone playing at or below a standard of Level 5 (ABRSM gradings) is still a beginner. At my current rate of progress I'll therefore be a beginner for another 6-12 years (gulp!). But learning piano is a fun thing to do, anyway. I won't stop just because I'm not a natural.