I've tried many times to learn piano, but get bored quickly and drop it. Until I found an easy method to get going quickly:
This is super fun, but where do I go from here? For example, I can play Piano Man using this tab, but breaking down the melody is a huge leap. What's the intermediate step?
To actually play the bass/accompaniment/melody simultaneously is, as you say, a huge leap.
A smaller, effective jump is to use what you already know, but swap it around a bit, to play this like Billy does. Each chord will have basically 3 notes. At the moment, you say you're playing the root with the left hand. That's fine, but if you split the chord notes, and use C bass on C, B bass on G, A bass on F, then G bass on the next C, etc., you'll be playing down the C scale with your left hand, but still using a note from each chord on the way. To make it sound even better, double up the left hand notes to play octaves. This means moving your whole left hand, instead of just fingers, but it sounds good.
The next step might be to keep the left hand, and only play the melody with your right. When this works - it'll sound quite bare - try putting a small chord under the tune notes, just on the beats as the underlying chords change. This should keep you busy for a while !!
Of course, if someone sings the song, you won't need to be playing the melody - that'll be done for you.
The 'next' step would depend on what you want to do next. What you are happy with.
I mean, if you just want to play chords to accompany other musicians that play with you, just keep practicing chords.
If you want to lead, you'll need to play melody. @Tim described pretty good some things on how to learn to play melodies, keep those in mind.
Also you would have to ask yourself what type of music you prefer. If you like classical, it would be good to get a beginner's book with some simple classical songs to practice. If you like jazz, you should do the same with some jazz books. If you want to play songs that are 'pop', what you are doing is good. Looking up their tabs.
Also, it would be good to learn some theory. That would help you understand why the song goes C - F - G - C or something.
That isn't really necessary if you don't want/ have time/ are bored or whatever, but it helps alot!
Although I also had piano lessons, what you've done with lead sheets is very like how I learned to play by ear.
Different people's brains work in different ways, so I can only describe what worked for me.
Start with simple tunes - Jingle Bells; Happy Birthday, etc.
If you want to develop your skills at working out chords, then start by working out the chord sequence. Otherwise, get the chords from a book or online.
Play the first chord with your left hand. Now play any note with your right hand.
Listen. Is it the right note? If so, good! Otherwise, listen again. Is it too high or too low? By a lot or a little? Use what you've heard to pick a different note to play with your right hand. Keep going until you find the right note.
Then do it all again with the second note.
Every now and again, play the notes you've worked out so far, in the right rhythm. When you reach a chord change, play the new chord with your left hand.
Pay attention to the way the notes fit in with the key you're in, and the chords you're playing. Simple tunes usually only use notes from the scale, so your choice of notes goes down from 12 per octave to 8. Some notes will match the chord you are playing, other notes will be outside the chord. Pay attention to which ones do and which ones don't.
By doing this lots - if your brain is like mine - you'll come to learn:
Once you are OK with simple songs, move on to more advanced ones. You will come to recognise what makes a tune simple or difficult. If you find that a particular tune is too hard, don't be afraid to put it aside until your skills have improved.
It's interesting to note that the answers so far relate to harmonic issues (ie which note to play), so I'll offer some different advice: try changing the rhythm of the tunes that you know. Play a rock song as a bossa nova (or vice versa)! Learn how to swing! Figure out how to play in 5/4 songs written in 4/4 time!
Going back to harmony, try changing the harmonies of a song - keep the tune but try out different chords.
If I remember correctly, jazz pianist John Dankworth recorded either a tune or an entire album called 'Adventures with mice', in which he basically took the tune of 'three blind mice' then played it with all manner of variations (I've never heard this, so I'm basing my statement on a description which I once read).
