Let's Play Kids Music provides a beginner version of Happy Birthday using only note names and fingering patterns. For the more advanced beginner, 8 Notes has a more complex arrangement of Happy Birthday, also for free, written on staff paper and using both right and left hands.
For a brand-new player using on the right hand, the most difficult part of Happy Birthday is the second last line, "Happy birthday dear person". This is due to "birth" being on a higher note, requiring the hand to move up the keyboard, coming back down to the first position by the end of the line. Practicing scales, broken triads, broken four-note chords, and arpeggios will help train the hand for that motion.
For players attempting the more advanced arrangement, the same line will also require more practice, due to the left hand playing more closely with the melody during that line, as well as the right hand still having to move up the keyboard. Players at this level will benefit from practicing the left and right hand separately and becoming comfortable with each individually before combining the hands. Beneficial exercises for the left hand for this piece include playing stacked triads or stacked four-note chords.