Let's start with reading. I think that reading music is a very simple skill, at least at the beginning. I start my students off with it in the first lesson. Oddly, no-one taught me to read music. My mother never learned and I played by ear for the first two or three years as a pop musician. When I realized that if you wanted to work with orchestral instruments, you needed to write down what you wanted them to play, I just taught myself.
Here is how it works: musical notes are written down with hollow and filled dots on a grid. Each dot stands for a single note. That is really all there is to it! Historically it took a long time to work it all out, especially how to show rhythms, but the system works very well. The grid is called a 'staff' and it has five lines:
That funny looking thing at the beginning, called the "treble clef", is a very stylized letter 'G' and where it curls around the second line from the bottom it shows that that line is the note we call 'G'. Right after it is a thing that looks like the letter 'C'. That is a "time signature" that tells you about the rhythmic structure of the music. It isn't a letter 'C', but a half-circle and a sign that dates back to the Middle Ages. It actually means "tempus imperfectus" and means, nowadays, that each little box, called a "measure" will have four beats. That hollow thingy sitting in the second space from the bottom is the note 'A'. Music, unlike the alphabet, only has seven notes: ABCDEFG. So if you go up from A, you go to B and C, but if you go down from A, you go to G and F. So that's why the note in the space above the line that we know is G is A instead of H or something.
I teach guitar, not piano or violin, so now I will show you how the staff relates to notes on the guitar. Here are the open strings of the guitar as they appear on the staff:
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As you can see, the notes on the guitar go far outside the staff! The reason is that what I am showing you here is a kind of notation originally developed for singers--vocal notation. So a five line staff is big enough to accommodate the range of most voices. The piano has a very large range, so they get to use two staves: the treble one that I've shown here, and another lower one that uses a bass clef instead of a treble clef. Guitarists manage to get by with just the one. OK, so if you sit down with your guitar and play from the bass strings (the thicker ones) to the treble strings, you will play the notes I showed above. All those notes on the frets that the left hand plays I'm just leaving out for now!
We have some notes, now we just need some rhythms. Here is what rhythms look like, written down:
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This is all played on the FIRST string of the guitar, that is, the thinnest, highest-sounding one. Count aloud a few sequences of ONE, TWO, THREE, FOUR. Make sure that you count evenly. What you are doing is just measuring time in packages of four. Four what? Four beats. Tap your foot if it helps. Ok, now you are ready to read the rhythms above. The first bar (up to the first barline) is easy. Count 1234 and every time you say a number, play the first string open. The next bar has one longer note. It takes up two beats: 1 2. As you say "1", play the note and let it keep ringing as you say "2". That's it. Then just play two more notes for beats 3 and 4. The next bar has two longer notes so you play on 1, let it ring as you say "2", then play on 3 and let it ring as you say "4". The last bar is easiest of all: play the note as you say "1" and just let it ring as you say "2" "3" and "4". If you have got each bar figured out, then go back and do the whole line. Congratulations, you can read both notes and rhythms. Everything else is just refinements!
Now am I oversimplifying? Well, I think I am simplifying just the right amount. Compared to reading words in a book, reading music is really pretty simple. Now of course, you spend your whole life learning the refinements and subtleties, just as you do with language. Why just the other day I learned a new verb: to impend, which I had known only as a gerund.
After all that, time for a treat. Notice that they are playing while reading the notes:
UPDATE: a friend sent me some good suggestions via email, so I have gone back and clarified some things.