Another idea that has more to do with biology than anything else, is to play the piece you are learning immediately before going to bed. It should be the last thing you do before laying your head on the pillow. During the day, our brain stores many things in short term memory. These memories are transferred to long term memory during sleep. If you are doing anything after your practice, the memories of your practice session will tend to get overwritten by the events you do later in the day. Playing the song one time just before bed, will bring these memories to the foreground. They will be transferred to long term memory more efficiently.
Keeping the Rhythm
The trick is to play what you play in the proper rhythm, even if you only play a portion of the song. In other words, practice keeping the rhythm going at the expense of everything else. Don't play the cords, or play one or two notes in the cord rather than the whole thing. Before long, you will be able to get more and more notes included in the cords, or you will learn which notes are optional. In most cases nobody but you will realize that there is a note missing.
Where to Put your fingers
The orthodox technique, for example if you are going up the scale with the right hand, would be to play the first three notes starting with your thumb, and then after playing the note under your middle finger to play the next note with your thumb again. (The thumb crosses under.) Going back down the scale, starting with the pinky on your right hand, you would play five notes, and then your middle finger would cross over the top of the thumb to play the next lower note.
Playing with the left hand is a mirror image of what the right hand does.
For regular music:
The same sort of arrangement applies, but it is also common to just pick up your hands and move them so that they span the next sequence of notes you are expecting to play.
Playing with Both Hands
(Nearly always) play with both hands, don't try to separate them.
Don't worry about skipping a note here or there in the harmony. If you don't get there in time, just go on to the next one, or make up a harmony! Few listeners will be able to tell. After a while your left hand will develop a feel for where the notes might be expected to be. After a while, your right hand will be able to pick out the melody essentially by ear without having to pay so much attention to the notes on the page, thus allowing you to pay more attention to the left hands harmonies. These two trends will combine together to make playing easier and easier. Doing a little bit of playing by ear, and a little bit of sight reading new music, will help to strengthen these trends.
Children and the Piano
So often, a parent will say, "Don't touch the piano", or "Quit making such awful noise on the piano", then the parent wonders why the children don't want to play the piano when formal lessons start. Children learn best by doing. Let them goof with the keys while you play. (It's less intrusive if you put them on the right side of you!) Or let them make horrible sounds by themselves, and if they accidentally play something that sounds good, be sure to acknowledge it, so it will be more likely to occur in the future. My son by the age of 8 was playing well both by reading music, and by ear, without ever having had a lesson. He just tinkered around. Not saying that there aren't some pretty unmusical things coming out of the piano, (even now), but overall the joy of listening to him play compensates for any bit of patience that was required.