What are the best ways to play with my 1-month-old?

Gentle touching. At 1 month, a baby is very young and doesn't do much playing. Eating and sleeping are the main activities. A parent can spend time talking to the baby, making eye contact when the baby is alert, cuddling and stroking the baby, or gently playing with the baby's hands and feet.
Talking, bright toys. At this age it is very important to talk lots to your baby so they begin to learn your voice. Put your face 8-10 inches away (your baby sees best at this distance) and smile and talk. Babies this age also like contrasting colors- light and dark, so use toys which have these.