In just one room in my house, the first water out of the cold water tap tastes moldy. What to do?

Please tell us the brand and the model number of your faucet, at the mean time remove the Aerator see the image below, turn on the water and see if the smell is still there if not clean the Aerator install and see.

To clean the Aerator put it in a glass and pure some vinegar soak it for a few minutes and clean it, good luck.

John