The complement "a" is used when the verb requires it: [ir a] Vamos a estudiar. ¿Quién va a pager? [empezar a]: Empezó a llover. A las seis empieza a oscurecer. [volver a]: Volvieron a elegir al mismo presidente. ¡No vuelvas a hacer eso!
Note that [ir a] and [venir a] can be used to express a purpose, but it is weak and general: Vino a ver a los niños - He came to see the children. Vino para ver a los niños - implies that he made a special trip, or that you consider it exceptional, or that he came to see the children and nothing more. For example, if he came to see the children and you asked him to fix a window, he would say: "Vine para ver a los niños, no para hacer reparos."
The particle "para" is used to express a strong or specific purpose: Vine para hablar contigo. I came (specifically) to talk to you (about something important). Vine a hablar contigo. (I don't know anyone else at the party, so) I came to talk with you (about nothing in particular). Jorge está guardando dinero para comprarse una casa. George is saving money in order to buy a house. Los esperó en el callejón para matarlos. He waited for them in the alley (with the intention) to kill them. Para hacer te, hay que hervir agua. In order to make tea, you have to boil water.
Finally, the pattern [estar al + infinitive] means that you're on the point of doing something: Estoy al llamar la policía. I'm about to call the police (if this continues). Estaba al salir por la puerta cuando sonó el teléfono. I was about to walk out of the door when the phone rang. The pattern [no estar para + infinitive (or noun)] expresses that you're not in any condition or mood to do something: (sick person) Yo no estoy para beber - vayan sin mí. I'm in no condition to be drinking - go without me. (parent) No estoy para juegos - dime la verdad. I'm in no mood for games - tell me the truth.