Understanding action reaction in an example When I move an object that object should move me as well. I tried standing on a skateboard to reduce friction, and holding a heavy barbell, move myself away from it (while still holding it) but I haven't noticed any movements of a skateboard. Am I missing something here?
 A: You probably just didn't throw around enough weight to overcome friction.  
While technically, if you were on a completely frictionless skateboard, by simply moving the barbell around you would roll a bit (the center of mass of you and the barbell would remain stationary), in the real world you have to overcome some friction to start rolling.  Friction is typically modeled as a fixed fraction of your weight:
$$ F_{\text{friction}} = \mu m g $$
where $\mu$ is the friction coefficient, and depends on the particulars of the materials or surfaces involved.
But, we can actually generate some quantitative estimates for whether you should have expected to start moving in this case.  By moving the barbell with your arms, you are causing a force to act on the barbell. By Newton's third law, this means there is an equal and opposite force acting on you.  This force, by Newton's first second law $ \vec F = m \vec a $, will be dependent on the mass of the barbell and how quickly you manage to accelerate it.
Assuming you weigh 75 kg (because its a nice number), and the barbell was 10 kg, if you threw your arms out from you, with a max speed of ~ 1 m/s at half an arm's length, you would have caused a force of magnitude:
$$ F_{\text{push}} = m_{\text{barbell}} a  = \frac{ m_{\text{barbell}} v_{\text{max}}^2 }{ 2 x_{\text{max}} }  \sim \frac{ (10 \text{ kg}) (1 \text{m/s})^2 }{ 2 (0.5 \text{ m}) } \sim 10 \text{ N} $$
Comparing this to your gravitational force:
$$ F_{\text{grav}} = m_{\text{you}} g \sim 750 \text{ N} $$
You would need the static rolling friction coefficient to be less than $10/750 \sim  0.01 $.  
It is unlikely that the friction coefficient is this small.
One way to see this, is if you were to put the skateboard on an incline by itself and raise the angle until it starts to roll, we could measure the static friction coefficient for your particular surface and skateboard.  Since the ratio of the tangential force to the gravitational force is $\sin \theta \sim \theta$, even without performing the experiment we can compute the angle a friction coefficient of 0.01 would correspond to, in order to get a sense of whether we would expect you to have moved.  $\theta = 0.013 \text{ radians} \sim 1\text{ degrees}$, which looks like this:

So, assuming the parameters above are correct, the fact that you did not move is no less mysterious than if a skateboard placed on that incline were not to move, which I personally don't find too mysterious. Especially if the surface was in any way rough.
In order to make this experiment feasible you're going to have to increase something by a factor of a few (~3), either moving around a few times more weight or moving it away from you a few times faster.
