Take your time. Songs don't have to be bright and unremittingly cheerful for children to enjoy them. Have a peaceful, restful moment sharing a lullaby. They work from day one. Here are some of my traditional favourites.
Lavender's blue dilly dilly
All the pretty little horses
But they're so easy to make up. My lyric memory is rubbish so I'm forever filling in the gaps with nonsense words or sounds. If in doubt, hum. Humming with baby's head tucked under my chin gets her ear pretty close to my voice box. I don't have proof that it works better (other than a sleepy baby), but can't help feeling it must have a bit of the vacuum cleaner effect - that low drone that babies seem to fall asleep to so easily.
And for a restful, waking moment, here's a little snail-paced rhyme to try for bigger babies and toddlers. Make snail shells with your closed fists and peeping horns with uncurled fingers.
Two snails sleeping, Two horns peeping, One snail creeping On one silver trail.
One snail sleeping,
Four horns peeping,
Two snails creeping
On two silver trails.
© 2002, 2015 Sheena Roberts