Here is a place Spanish does want the -ando or -iendo form: after seguir, to mean carrying on doing something.
sigo hablando = I continue speaking
Sigue hablando · Es probable que siga hablando · Es imperativo que siga siendo gratis
Siendo is the -iendo form of ser, so continues being is sigue siendo.
You know verbs of movement take a connecting a. Beyond those, only a small number of verbs need a preposition before a following "to" form, and each has its own.
Es espectacular que comiencen a hablar español · Cuando terminen de venderlos
This is the useful generalisation. Where a verb joins a "to" form with a preposition, it can usually also take the -ing form instead, with a different meaning.
English barely distinguishes begin to speak from begin speaking. Spanish does.
Quiero que hablen conmigo cuando terminen de venderlos · Quiero que hablen conmigo cuando terminen vendiéndolos
Worth noticing inside these longer sentences: cuando terminen takes the mood tense because it refers to a future event that has not happened yet.
When cuando means generally, the plain present is right.
Gratis is where English gratuitous comes from: unnecessary, extra, free. And espectacular needs its e on the front, since Spanish cannot begin with s plus a consonant.
The teacher's closing advice on this whole topic, and it is practical.
What the course set out to teach here is the idea of the mood tense, its general feeling, and the constructions it appears in. The rest comes from listening out for how people around you use it.