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.