Everywhere states that a MOSFET has no current at the gate. However, this is not true since, during the time the gate capacitance charges up to reach the certain voltage threshold, the current is entering the gate capacitance.
My actual question:
During the charge-up time of the "gate capacitance", does the current only go to the source? (Meaning current enters the gate capacitance while charging and then leaves through the source for a N channel type?)
Remember, there are two (major) gate capacitances in a power MOSFET: the gate-source capacitance Cgs, and the gate-drain capacitance, Cgd. If the drain voltage never changes, then yes, all gate current goes into the gate-source capacitance.
However, in power MOSFETs, the drain voltage almost always changes. During turn-on, at the point of switching, the drain voltage in an NFET will start to fall. This discharging of Cgd causes current to flow from the gate to the drain. Depending on drain-source voltage Vds, transistor and driver parameters, the gate-source voltage Vgs may have a small inflection point, a Miller plateau, or wild oscillation.
For a better discussion on the Miller plateau, see this answer