If the character isn't on the ground, check if the collider (e.g. character controller) is set up correctly in the running scene. Most likely the collider is on the ground, but placed too low :)
The movement animations are played automatically, if you've set it up that way (there's a setting for that in the combatant settings). If you're using mecanim, there's a tutorial on that in the gameplay tutorial section, or take a look at cat's tutorial.