Often, those buttons allow you to shift out of reverse. If the button and/or linkage are damaged it may not do anything now.
Some vehicles have a button on the gearshift that controls the shifting between economy mode and performance mode. Economy typically shifts sooner, performance gives you more time in lower gears so that you can accellerate faster.
On my '96, the transmission release button is the big one on the front side of the shifter. You have to pull it toward the stick to take the car out of Park. You also have to be putting pressure on the brake pedal at the same time. (Danged government regulations!)
On some American cars, that switch on the SIDE of the shifter, is designed to keep the transmission
from shifting into overdrive. It holds the transmission in 3rd gear. You would
want to use this primarily on hills or grades when the transmission is "hunting" IE;
shifting up and down between 3rd and 4th (overdrive)
Admittedly, I've seen this mostly on Fords, but it might be the case here.
If nothing else, drop into a GM dealership and ask the service department folks.