What are the best ways to play with my 3-month-old?

Explore sensations. At 3 months old, he has started to interact with his environment and is much more aware of it, it is a great oportunity to explore the outdoors with all it's colors, sounds and sensations. As he is now starting to communicate and expres more emotions through cooing, smiling, giggling, and different cries use this to expand his sensations by reading to him. Practice reaching and tummy time.
Make the baby smile. Three to six month old babies like to play. Activities that make a baby smile are good to continue, such as holding/cuddling the baby, gently teasing the baby with a stuffed toy, singing a song to the baby, talking to the baby, a brief baby massage, playing peek-a-boo, etc.