To complete the badges' missions, you must first spot them. Once you see the badge mission, shake the remote and you will complete the badge mission.
To see which badges you have, select the location you wish to check and swipe over to the left to highlight the PAW print. Click on the touch pad and your badge display will open.
If you wish to know how to complete a badge, open the badge display for the location. Swipe to the badge you are interested in and click on the touch pad.