symbolipoint said:
So, people here who understand, tell us about left & right hands coordination with playing piano.
Please note that I'm answering from my personal perspective below. If I would categorize piano players into basic, average and advanced, I would consider myself to be an average piano player.
What I mean by that is that I can pretty easily do basic accompaniment of a played piece. If I have to learn a melody I would have to practice some. I were to play "fully", i.e. melody with right hand and chord patterns and/or bass patterns with left hand, I would have to put in quite some practice depending on the difficulty of the piece.
symbolipoint said:
Plenty of piano students and piano players seem to demonstratably not have trouble with coordination of the hands either, but I do wonder: Is it really so easy, really so natural, (piano) [...]
I would say no. It's not easy, particularly in the beginning

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(Edit start)
I could be more specific here (again from my own perspective). In the very beginning, I would say keyboards and piano are more easy because the effort to play one note requires only one hand. On a guitar (unless it's an open note, i.e. a string played without placing your hand on any of the frets) one note requires two hands.
But piano and keyboards then get sort of incrementally difficult, so to say, depending on the difficulty of pieces and styles of music. But so does the guitar. It's sort of, how can I say it, climbing a mountain with hills in between.
When you start playing your first instrument the very first time, you are standing at the bottom of a mountain, and if you continue you have the intention of going up.
But you don't exactly climb this mountain from bottom to top. You climb up individual hills, and as time goes by the hills get higher (the pieces get more difficult), but your ability to climb has also increased. And the higher up you are on the entire journey, the better/more experienced climber you are, i.e. better/more experienced player). It's my way of describing it

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(Edit end)
symbolipoint said:
Left hand plays one type of part and right hand plays another type of part. Fingers of both hands press to play on keys. Guitar and violin are not like that to play. One hand handles valving while other hand strikes the strings.
You are correct. But first I have to point out that violin and guitar are also different from eachother:
Violin is one of the most difficult instruments there are, as far as I know.
A big difference between violins and guitars is that the fingerboard of guitars are (normally) fretted, while the violin is fretless (though there are fretless guitars and basses too). Furthermore, the fingerboard of a violin is really small, which makes it even more difficult for the fingers to achieve the right pitch for a note (smaller differences in length along the strings for different notes, so to say).
Another thing I want to point out is that guitar can be played in different ways, e.g.
1) Single chord (playing just a single chord at each chord change)
2) Strumming ("playing a chord continously" while keeping the rhythm, and then changing the chord as the song goes on)
3) Arpeggiated playing ("filling out" each chord by playing different note patterns within each chord)
Coordination between left and right hand increases in difficulty here from (1) to (2) to (3).
Here's a video where the superb guitar player Mark Knopfler shows different techniques:
(please note that he is exceptionally good, so what he shows here is not for beginners

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Mark Knopfler on Guitars
symbolipoint said:
Then further, I am figuring that bass note pattern (left hand) for piano and other parts done with right hand would seem too difficult to acquire to be able to do, yet plenty of piano players seem to show no trouble.
They have practiced. A LOT

. And on many different piano pieces.
pinball1970 said:
I think I learned chords first with my right hand then put bass notes in but simple, C chord C in the bass. Then adding more notes as I became more confident. I do not remember struggling with the co-ordination at that time. it was playing the right note with the right finger that got me (and still does)
I think that is the way to go in the beginning.