the shifters are indexed, and function no differently than twist grip shifters. They pivot about the round features at the bottom of the levers.
The peddles need to be going around to shift the gears.
Given that it is a Wal-mart bike, you should consider taking the bike to a bike shop to make sure that the derailleurs, brakes and cables are adjusted properly. Ditto on the headset and bottom bearings, spokes and wheel bearings. This is going to cost $$$.
Have someone hold the back wheel in the air and turn the peddles while you move the levers to see what they do.
Given your lack of knowledge about bicycles, I strongly suggest that you return the bike to Wal-mart and purchase a bike from a bike shop that will teach you about the bike and service it for you - overall a bike shop bike will cost you less than a Wal-mart bike - assuming that you ride it and not just stick in the back of the garage.