At the Merchant
  1. Insert Card in the Merchant Terminal
  2. Leave the Card in the Terminal
  3. Sign Receipt or Enter a PIN
  4. Remove your Card.

The chip card still has a magnetic strip, so you can still swipe and sign, as well.

At the ATM: Use the new "chip" card the same way you would use your old card. Insert it into the ATM machine, enter pin and complete transaction.