There are 3 primary ways to kick a footbag: the toe kick, the inside kick, and the outside kick. You can also hit the footbag with your knee and head.
The Toe Kick
To do a toe kick, bend your knee and bring your leg up while kicking with the top of your foot. This will give you much more control of the footbag when kicking.
The Inside Kick
The inside kick is one of the fundamental footbag kicks. Bend your leg upwards and kick the footbag with the inside of your foot. Make sure to kick all the way up. You want the flat side of your foot to be level to the floor.
The Outside Kick
The outside kick uses the outer surface of your foot. Bend your leg away from you and kick the footbag with the outer side of your foot. This kick is usually used to bring the footbag back in front of you, but it is also used in tricks.
There are some helpful ways to kick that make it easier to keep the footbag in play. Generally a good way to kick is with the inner side of your foot. Check out this great explanation from Elemental Footbags founder, Lotus: