Quote by sgstudent
Since the force that we exert is smaller than the surface area of the piston
Thanks for the help guys!

How can a force be less than an area?
Perhaps you mean the force you exert on the first piston is exerted over a smaller area than the area of the piston against the fluid? Irrelevant.
You somehow exert a force F1 on the first piston.
That piston spreads the force over an area A of fluid, generating pressure P = F1/A.
The pressure is transferred throughout the fluid.
At the second piston the pressure acts over area B, generating force F2 = P*B = F1*B/A.
Regarding pressure at depth, consider a small cube, area C on each face.
Force on each face F = C*P.
As C tends to zero, F tends to 0, not infinity.