I pretty much agree with what has been said here, but I can add some additional information.
I started learning how to play keyboard on organ. On an organ, you can play a chord with your left hand, and it will sound as long as you hold down the notes. Additionally, as most organs have at least two keyboards, you can play that left-hand chord in a register where it sounds nice and clear (instead of low and muddy). And then, you add bass using the bass pedals.
Obviously, you have to modify this technique when playing on a piano. You cannot just hold down a chord on a piano, because the volume of the notes will fade out. And you usually would be playing those chords in a lower register on a piano, where the sound would be too muddy for a regular chord. So, instead of a regular chord, I might play an octave (such as a C, and the C above it), or octave-fifth-octave (C, G, C). And then, to give the piece rhythm, I might play those bass notes in quarter-note rhythm -- or maybe eight-note -- or maybe something else, depending on the type of song. Or, as suggested above, I might arpeggiate the notes -- continuing the C-chord example, playing something like C-G-C (1 octave up)-G-C, or C, E, G, C (1 octave up), or maybe C-C (1 octave up)-G-C (1 octave up), or perhaps some other pattern.
But then, you still need to fill in the harmonies of whatever chord is called for in the song, so for that, I play right-hand chords for the melody notes that fill in these extra notes. Normally, the highest note of the right-hand chord would be the melody, so in a simple example where the song calls for a C major chord and the melody has an E note, I would play a C chord with an E as the top note -- such as G, C, E (lowest to highest notes). Of course, not all melody notes will be part of the chord, so for that, you just have to come up with what you feel sounds good. Maybe only play the right hand chords for the melody notes that are part of the chord. Or maybe just "bend" the chord to fit the melody, so that, for example, if the melody note is an F, and the background chord is C major, you could play G, C, F (lowest to highest notes) for that note. That is actually a C sus4 chord, by the way.
But another useful thing to do, especially when looking at what to do with your left hand, is to listen to the original recording of the song, and maybe check the sheet music for it, if you can find it. Even if you don't play exactly what is played in the song, it can definitely give you an idea of what kind of rhythm and notes to play with your left hand throughout the song. For example, in the Vangelis song "Chariots of Fire", I play a repeating Db sixteenth-note with my left hand at the beginning and throughout what would be considered the "verse" of the song -- often holding down a low Db for maybe a quarter-note with my left-hand pinky finger while playing the sixteenth-notes on a higher Db with my thumb. But when I get to the "chorus" of the song, I change that to match the overall feel of that part of the song, often just playing the left hand notes once as the chords change, since the chord changes are fairly quick at that part of the song.
And finally, I definitely use the sustain pedal to help blend all this together -- but you have to be careful not to over-use it, otherwise, you get everything kind of mushed together. At the very least, you usually want to release and re-apply the pedal at chord changes.
