I would add that learning the treble side first, then slowing way down and adding the bass is the way I did it. Too much going on the treble side to memorize before trying to add the bass in the manner I described.
My two cents on working the bass side in with the treble side.
Learning the two hands separately will not get you there, not very soon. Your brain still has to learn to coordinate them together. Your hands are not naturally wired in your brain to do two different things at the same time. Our nervous system is wired symmetrically. You have to train yourself to break that.
What I found helped me is to slow the songs down SO SLOW, that your mind can begin to think of what each hand is supposed to be doing, and when. It will be really slow to get there, and won\'t be much fun.
As you start playing to that slow sound (I rely on hearing it in your head), with both hands, you can start to integrate it together into your head.
Again, it won\'t be fun. It will be work.
To what nunya says, I would add that when I come to a measure that is particularly difficult to coordinate between two hands, I'll practice that measure by itself over and over, maybe 50-100 times, until muscle memory really begins to set it.