There are some verbs that include and some that require the preposition a; without the preposition the message implied would be different. Let's notice the difference in meaning with and without the preposition a:

aprender a- to learn how to
Raquel aprendió a hacer las multiplicaciones con su padre.
Rachel learned how to multiply with her father.
Raquel aprendió las tablas de multiplicar con su padre.
Raquel learn the multiplication tables with her father.

comenzar a (ie)- to begin to, to start to
La multitud comenzó a gritar al escuchar la explosión.
The crowd started screaming when they heard the explosion.
Julio comenzó la tarea hasta las 8 de la noche.
Julio started doing his homework until 8pm.

contribuir a- to contribute to
El ruido contribuyó al movimiento de las ventanas.
The noise contributed to the window's movement.
El Senador contribuyó con la compra de los libros de la escuela.
The Senator contributed to the purchase of the school's books.

correr a- to run to
El cachorro corre al gatito.- to run to
The puppy chases the kitty.
La estudiante corre a la estación del bus.- to run to
The student runs to the bus station.
El famoso atleta corre los 100 metros.
The famous athlete runs the 100 meters.

empezar a (ie)- to begin to
Melissa empezó a comer antes que llegaran los invitados.
Melissa started eating before the guests arrived.
Melissa empezó su tarea hasta muy noche.
Melissa started her homework until very late at night.

enseñar a- to teach how to
Mi madre me enseñó a manejar bicicleta.
My mother taught me how to ride a bycicle.
Mi madre me enseñó las fotos de su boda.
My mother showed me her wedding pictures.

llegar a- to come to
El doctor llegará al hospital a la medianoche.
The doctor will arrive to the hospital at midnight.
El doctor llegó muy rápido.
The doctor arrived very fast.

volver a (ue)- to do something again
Francisco volvió a tomar alcohol.
Francisco drank alcohol again.
Francisco volvió de su trabajo.
Francisco returned from his job.

salir a- to go out to
Mi cuñado sale a correr todos los días.
My brother-in-law goes out jogging every day.
Mi cuñado salió del hospital ayer.
My brother-in-law left the hospital yesterday.

some verbs require the preposition "a" in a reflexive form. (if non-reflexive the meaning usually changes).

decidirse a- to decide to
Ella se decidió a casarse.- to decide to
She decided to get married.
Ella decidió comerse todo el pastel.- to decide to
She decided to eat the whole cake.

echarse a- to start to
El ladrón se echó a correr al ver a los policías.
The thief started to run when he saw the police officers.
echar- to pour
El dueño del restaurante echó mucho queso en la pizza.
The restaurant's owner pour a lot of cheese on the pizza.

acostumbrarse a- to become accustomed or used to
Mi bebé se acostumbró a que yo le cantara canciones antes de dormir.
My baby got used to me singing songs before going to sleep.

atreverse a- to dare to
Los niños se atrevieron a tirarse a la piscina a los 3 años.
The kids dared to jump into the pool at 3 years-old.

dedicarse a- to dedicate oneself to
Rodrigo se dedica a las ventas.
Rodrigo works on sales.
dedicar- to dedicate
Rodrigo dedicó una bella canción a su novia.
Rodrigo dedicated a beautiful song to his girlfriend.

negarse a (ie)- to refuse to
Romelia se negó a devolverle el anillo.
Romelia refused to give him the ring back.
negar- to deny
La Embajada le negó la visa.
The Embassy denied him the visa.

ponerse a- to start to, to set about to
Mi esposa se puso a cocinar para toda la familia.
My wife started to cook for the whole family.
ponerse- to turn
Mi esposa se puso triste porque no llegó su mamá.
My wife turned sad because her mom did not arrive.
poner- to place
Mi esposa puso las enchiladas en la mesa.
My wife placed the enchiladas on the table.

Some verbs always require the preposition "a".

aspirar a- to aspire to
Rita aspira a ser modelo.
Rita aspires to become a model.

ayudar a- to help to
Los bomberos ayudaron a la muchacha a salir del edificio.
The firefighters helped the girl to leave the building.

invitar a- to invite to
Mi suegra me invitó a almorzar.
My mother-in-law invited me over to have lunch.

ir a- to go to, to be going to
Vamos al mercado.
We are going to the market.

Note! The reflexive form is followed by a preposition plus
an infinitive, but the non-reflexive form is followed by an
infinitive without a preposition.