Hola Emanuel
What we mean in this sentence that you saw in this lesson is different to "empezar a leer".
Hoy vamos a aprender sobre la Revolución Francesa. Para ello, vamos a empezar leyendo sobre la sociedad francesa de esa época.
Today we are going to learn about the French Revolution. To achieve that, we will start reading about French society back then.
What we mean here when we use "vamos a empezar leyendo..." is:
We are going to start (to achieve this) by reading ........ This use of gerund is specific. It is expressing "by doing something" [you will achieve something]
There is a lesson on this specific use of the gerund.
Have a look here.
I hope it helps
Inma
