I originally learned to play keyboard on a home organ and took lessons for about 3 years, but was also taking piano courses at school. I agree that much of what I do on a piano -- or electronic keyboard, for that matter -- involves mainly improvisation for my left hand. I rarely look at full sheet music and play exactly what is written. I'll usually learn a song by ear or a "fake book" lead sheet, and then use my left hand to come up with an accompaniment or bass line that sounds more or less like what is in the original song. And while I do try to keep an accompaniment or bass line going with my left hand, I certainly cannot play like advanced Classical music players who can play sixteenth notes with both hands at the same time!
So, here is my advice...
First, if you don't already know them, learn the chords. At least get familiar with the major, minor, 7th, and minor 7th chords, and then eventually add others, like major 7th, diminished, and diminished 7th, before going on to the crazy jazz chords. But at least start out with the major, minor, 7th, and minor 7th. This is one of the first things I learned when I started to learn organ. Learn them in root position, then as you get familiar with them and used to playing by "feel" (such as a chord being every other white key, or two white keys with a black key in the middle, etc.), learn the notes so you can easily play inversions of the chords.
Now, as you know, unlike an organ, you cannot just hold down a chord on a piano, as it will fade out in short order. So, you can try different improvised backgrounds based on the chord, depending on the song. For example, maybe tap the chord in quarter notes with your left hand as you play the melody with your right. Or, as previously suggested, break up the chord into a mini arpeggio, such as taking a C major chord background (which has the notes C, E, and G in root position), and repeatedly play C, E, G, E in quarter notes with your left hand. Or, something I believe is called "Alberti bass", C, G, E, G (lowest note, highest note, middle note, highest note). Then when the chord of the song changes, change the chord or notes you play with your left hand.
Of course, a piano does not have bass pedals, so if you just play left hand chords and right hand melody, you might find that if you play the chords in an octave where they sound clear, you don't have much bass tone to your music, but if you play the chords an octave lower, they sound "muddy." So now, you can go to the method mentioned by DerekA earlier and play octave notes for the bass with your left hand. Start out with just the chord's root notes, as DerekA said, but then, depending on the song, branch out. Think of the Classic rock song "A Whiter Shade of Pale", with its slowly descending bass notes. You can play those bass notes, as octaves, while you play the melody.
Now, adding to these left hand techniques, you will likely find that when you are playing single notes, or even octaves, with your left hand, that if you just play a single note for the melody, the overall sound will be kind of thin and lacking some of the harmonies of the chord that the song calls for. This is where you can now introduce playing right hand chords that follow the melody. This takes practice, but generally involves playing a combination of notes with your right hand where the top note is the melody note, and then you use other fingers of your right hand to play 1-3 other notes of the background chord called for in the song. Keep in mind that the melody note will often not be part of the notes of the chord listed for the background at that particular measure. As an example, a song may say that the background chord is "C major", but the melody might be something like C, B, A, D, C -- or whatever the song's melody is. In this case you would play those melody notes as the top note of your right-hand cluster of notes, and then use other fingers to play some other notes of the C major chord. So, for those example melody notes (C, B, A, D, C), you could play the E and G below those notes with other fingers while your pinky or ring finger plays the melody notes. Then, if you are playing octave C's with your left hand, you now have a pretty full-sounding tune going! And remember, if you're playing octaves, you can play them in rhythm so that your left-hand sound doesn't fade out by just holding down the notes on the piano keyboard.
Finally, there's that sustain pedal. Of course, as I'm sure you know, that pedal allows notes to ring on after you let go of the keys. This can be a great way to get a full, lush sound, but when overused, can make everything sound "blurred together" and even dissonant. So, in most cases, if you are using the sustain pedal, you will want to quickly release it and then step on it again when the background chord of the song changes. And when should you use the sustain pedal? I would say, if you're playing a song that has quick, punchy notes, such as maybe a dance or jazz song, you'd want to limit your use of it. But if you're playing something with full, lush backgrounds, such as many songs by Enya, or the "Chariots of Fire" theme by Vangelis, then you'd want to use the pedal, but again -- releasing it and re-applying it between chord changes.
One other thing I do is that, when learning a new song, I do not use the drum rhythm background at all. I'm first trying to learn the notes and chords, while also slowly learning the beat/rhythm of the song itself, and I don't want the added distraction of trying to keep up with a preprogrammed drumbeat at that point. Once I get comfortable with playing the song and it's notes, chords, and rhythm, then I add the drumbeat and "fine tune" the rhythm of my playing of the song.
I hope this helps. I know it is a lot, but the important thing is to practice and take it a little bit every day, starting with learning the chords, then using the chords and playing them in tempo or as mini arpeggios with your left hand, then trying bass octaves while adding notes of the background chord to your right hand.
And remember, you can mix and match different playing techniques throughout a song, depending on the tone and character of the different parts of the song. Think of Enya's "Orinico Flow". Throughout most of the song, there are strong beats, backgrounds, and bass -- perfect for playing octave bass notes with your left hand while playing right-hand note clusters for the melody. But in the middle, there is a slow, soft part -- Here, you might want to just play gentle chords with your left hand -- in an octave where they don't sound "muddy" -- and not worry about the bass too much, because there isn't too much bass in the original song at that point.