What a verbal periphrasis is
A verbal periphrasis is a fixed pairing of a conjugated helper verb with a second verb in the infinitive or the gerundio. You already know some, such as ir a + infinitive, estar + gerundio and tener que + infinitive. Here come more, for the beginning, the end, the continuation or the repetition of an action. English has the same idea (keep working, stop smoking); the work in Spanish is the shape: which linking word comes in the middle, and whether the second verb is an infinitive or an -ando form.
hay que, it is necessary
tener que says who has to do something. hay que + infinitive has no subject and never changes: it states a general necessity, the English you have to in the sense of one has to. If you mean a specific person, switch to tener que.
Tengo que trabajar más.
I have to work harder.
a specific person
Hay que trabajar más.
You have to work harder.
in general, no subject
Hay que tener paciencia.
You have to be patient.
in general, no subject
The phases of an action
Beginning
empezar a / comenzar a + infinitive: start doing something. ponerse a + infinitive: get going, set about it.
Empecé a estudiar español.
I started learning Spanish.
Se puso a llorar.
He started crying.
Ending
dejar de + infinitive: stop doing. terminar de + infinitive: finish doing. Both English translations end in -ing, but the Spanish verb after de is an infinitive.
Dejó de fumar hace un año.
He stopped smoking a year ago.
Terminé de comer.
I finished eating.
Continuing
seguir / continuar + gerundio: something goes on. This is the one group where an infinitive is wrong: the second verb ends in -ando/-iendo. Think keep working, not continue to work.
Sigue trabajando muy duro.
He is still working very hard.
Continúa estudiando.
She is still studying.
Repeating
volver a + infinitive: do something again. English puts again at the end of the sentence; Spanish moves it into the verb, word for word return to doing it.
Vuelve a intentarlo.
Try it again.
Volvió a llamar.
He called again.
acabar de, the immediate past
acabar de + infinitive describes something that happened a moment ago, the Spanish equivalent of have just done. English marks just with a perfect, Spanish with a present: Acabo de llegar. The helper verb stays in the present tense even though the action is over.
Acabo de llegar.
I have just arrived.
Acaba de salir.
She has just left.
Acaban de llamar.
They have just called.