is there a way to find out if a player is moving?

Use the Humanoid.Running event. This event fires both when the Humanoid starts and stops running, so you need to check the speed to ensure the Humanoid is actually moving.

Taken from the wiki:

game.Workspace.Player.Humanoid.Running:connect(function(speed)
    if speed > 0 then
        print("Player is running")
    else
        print("Player has stopped")
    end
end)

Actually, yes. Humanoids have a "Running" event that triggers whenever they start moving. You can use that to detect movement, as seen below.

game.Players.LocalPlayer.Character.Humanoid.Running:connect(function()
    print("the player is moving!")
end)