What does 'Oppose a Change in Current' really mean from Lenz Law? We all know what Lenz Law is, but I have a bit of trouble conceptualizing the phrase above. Does 'Oppose a Change in Current' means it will take more time for the current to increase to its maximum value(let's say 5Amps)? Or will it end with less amount of current from expected(5A) to let's say 4A at the end?
If it opposes a change in current, does that mean extra voltage/energy is needed to drive/push this current due to this opposition? 
 A: So first, I think you mean to say that an EMF is produced that opposes a change in magnetic flux. Depending on the system this could also mean a change in current, but this is not always the case.
Second, this not as precise statement is really just just to determine the "direction" of the induced EMF. To get more information about the dynamics of the system you need to know more about the specific system in question as well as use more in-depth "machinery" in determining the dynamics (such as Maxwell's equations).
It is worth noting that just because an EMF is induced that opposes the change does not necessarily mean it completely overcomes that change. 
A: 
Does 'Oppose a Change in Current' means it will take more time for the current to increase to its maximum value(let's say 5Amps)?  

Yes.  

Or will it end with less amount of current from expected(5A) to let's say 4A at the end?  

No, as this would imply that the induced emf “wins” and the induced emf has stopped the change producing it but if that has happened, ie there is no change, there can be no induced emf as the emf is induced due to the current changing.  

If it opposes a change in current, does that mean extra voltage/energy is needed to drive/push this current due to this opposition?  

Energy needs to be supplied.
For example if a dc supply is connected to an inductor as the current through the inductor increases, the energy stored in the magnetic field of the inductor increases and that energy comes from the dc power supply.
That increase in the energy stored in the magnetic field of the inductor cannot happen instantaneously.
