Perhaps an easy way to see where Newton's second law comes from is as follows.
Imagine an object sitting in space (so no friction, etc. to worry about). You can push on it (exert some force on it), and see what happens to it. When you push on it with a constant force, you see the object start to accelerate, which means its velocity increases linearly with time. If the object is moving at 10 m/s after 1 second, then it will be moving at 20 m/s after 2 seconds. Now you try pushing on the object twice as hard, and you notice it accelerates twice as fast. If you push on it five times as hard, it accelerates five times as fast. It quickly becomes obvious to you that force and acceleration have a linear relationship, so if you multiply the force by some constant $\alpha$, the acceleration gets multiplied by the same value. Furthermore, since the object doesn't accelerate at all when you aren't exerting any force on it, you know that the two quantities (force and acceleration) must be related by an equation of the form $F=\kappa a$, where $\kappa$ is some constant. We call this constant mass, and denote it by $m$, but mass (or at least inertial mass, the kind that shows up in Newton's second law) is really just a measure of how much an object accelerates when you exert a force on it.