Practice these three chords: D, A7 and G. The chords are all played in the first position of the guitar, closest to the tuning keys. These are basic chords found in any beginning guitar chord book or online at no charge.
Strum these chords with a waltz feel. "Happy Birthday to You" is written in a 3/4 time signature, which means there are three beats per measure. Don't over think the rhythm. Just simply strum up and down and count (1 and, 2 and, 3 and) as you play. Repeat strumming with this count while you play the chord progression in step 1. This will help you get a waltz feel and a good rhythm for singing along to the song.
Start on D and play a measure and a half intro (count 1 and, 2 and, 3 and, 1 and, 2 and), then come in singing the first line. Change to the A7 chord on the word "you" at the end of the first line; change back to the D chord on the next "you."
Play the G chord when you sing the person's name in the song, then on the word "birthday" after the name, play a D chord. Play A7 on the "to" and end on a D chord on the last "you" in the song. To give the song a little color, you can strum the last two chords more slowly and let them ring out as you hold out the words when you sing them.