At three months, the cognitive changes are dramatic enough that you'll notice them without looking. Your baby is interested in everything — studying faces with new intensity, tracking objects across the room, craning their neck to see what's happening behind them. Cause and effect is clicking: put a rattle in your baby's hand, they shake it, notice the sound, and shake it again. That's the beginning of goal-directed behavior.
Social smiling is constant. Your baby smiles at familiar people, at strangers, at themselves in the mirror. According to Zero to Three, this abundant social smiling indicates healthy emotional development. They're also reading your emotions — research shows babies as young as three months respond differently to happy, sad, and angry facial expressions.
Cooing is evolving into babbling — you might hear "goo," "gah," or "bah" mixed in with the standard coos. The serve-and-return exchanges are getting more elaborate, sometimes going back and forth for minutes. Crying has decreased significantly from the 6-week peak — you can both breathe easier.