You know how if you have an object and you put a force on it, its not the whole object that moves at the same time, but instead, the force pushes the object and goes through it at the speed of sound? Well, what is this like when you only push a particle, such as an electron? Does it travel through the electron at the speed of light?

A particle is a point-like object, so your question is ill posed. Particles never actually make contact like that, they interact via fields. This given them a effective size. Anyway, and small changes in the fields will propagate with a speed limited by the speed of light.

It is only 'contact' forces which propagate through an object at the speed of sound ( although sometimes supersonically ), such as when you hit a baseball with a bat.

Forces due to a field, such as gravity, EM, etc., are 'instantly on both sides of the object, because the field is 'already there'.

Changes in the field propagate at c, as AJB has stated.