the main factor here is that ping pong balls are really light weight, so the impulse created is very small. So most of the time the spin on the ball will be about the same whether you hit it already spinning or from stationary. However the angle it leaves the bat does change quite significantly. If someone does a really powerful backspin shot, the best way to get it back is to do the exact same shot with the bat angled backwards, this reverses the spin and so will cancel out the effect of the spin.
If you try to top spin a back spin shot, then it will nearly always go into the net due to the angle of your bat and the effect of the backspin, so you have to hit it upwards much more as the backsipn makes it want to travel into the net.
So the rules are;
1. to return a backspin shot with backspin, hit it back with the same amount of backspin, as it cancels the spin.
2. to return a backspin shot with topspin, aim a lot higher than usual and with more topspin
3. to return a topspin shot with topspin, hit it with the same amount of top as they did to cancel the spin.
4. to return a topspin shot with backspin, aim much lower than usual and with more backspin
And the best type of spin to put on is sidespin, as the opponent always has to think about which side the spin is going to make the ball go. Especially when serving, and especially if you can hide the direction you hit it by shielding it behind your shoulder like I used to do. Most players don't use sidespin in games as it takes longer to do the shot and table tennis is a very quick game at pro level, pro's usually stick to top and backspin.
When it is a backspin or topspin, it is easy to work out, you apply the same spin as them to make it easy, or if your being adventurous you can alternate it, but this requires you adjust where you hit the ball quite severely.
and according to them, topspin on a topspin generates more spin than topspin on a backspin...here, check this out.
I think that's wrong. It may feel as if you are generating more spin when you hit it, but that is because you are putting more force into change the spin of the ball back the other way. When its a slice shot onto a topspin you are effectively re-enforcing the spin already on the ball.
Saying all that, it depends on what your bat is like. The one I used to use in tournaments cost about $250 (yes, rip off

) as it had amazingly sticky grip and the ball would never slide across the surface. But some people choose to use bats with no grip at all, so the spin deosn't effect them as much. So it totally depends on the 'stickyness' of your bat really.
THIS table tennis match is a perfect example of how to alternate top spin to back spin and back. One person is brilliant at top spin, the other at back spin. And i mean
really brilliant, one of the best ping pong rallys I've seen.