PCB design, Ground plane and power plane issues
My understanding is, Ground plane is full of copper unless I route any signal through them.


If you have a ground plane, do not route tracks through it. That defeats the object of having a ground plane. It's OK for the ground plane to have holes where signal/power vias go through, but don't let the holes join up into a slot.

Vias to the unbroken ground plane will be OK. But be aware that a ground plane is not zero resistance. It's real copper, and has a small and finite resistance, and is extended so has a finite inductance, so currents flowing through it can generate voltages. Lay your sensitive components out so they they don't see the voltages generated by heavy or rapidly changing currents. This often sort of works out OK with a reasonable floor plan (analogue here, power there, logic over there), but it's still too easy to mess it up. If you let holes line up to make slots, or route signal tracks through your ground plane, this increases its inductance and resistance dramatically, and increases interference from high current to sensitive regions, so don't.

If you only have one power rail, and only logic ICs, then a power plane makes sense. In a mixed signal board, it tends not to work to your advantage, as even if you don't have several rails, you will probably want several different distribution regions, a 'quiet' rail for your analogue and 'noisy' rails for logic or oscillators, each individually filtered. Route your power with tracks, by all means mostly on layer 3, but don't get too stressed about putting power and signals on all three non-ground layers. 

Did I mention not to run tracks through your ground plane?