Si, puedes utilizar la construcción "empezar + a + infinitivo" en español, algunos ejemplos:
"Ya empecé a estudiar para el examen." ( I already started studying for the exam. )
"Empieza a ahorrar ahora, o lo lamentarás." ( Start saving now, or you'll regret it. )
Si, también puedes utilizar las construcciones "terminar + de + infinitivo" y "acabar + de + infinitivo", otro ejemplos:
"Ya terminaste de comer?" ( Have you finished eating yet? )
"El trabajador acabó de pintar la casa." ( The worker finished painting the house. )
Some final comments:
"terminar" and "acabar" usually mean the same thing, which is "to finish". But there are some expressions where only one or the other are used. For example, if you say "They broke up (as in their romantic relationship)." you would usually say "Ellos terminaron ( o rompieron )." In this case I don't recall people using "Ellos acabaron." You would be understood, but it's more common to say "terminaron". You could say, though, "Ellos acabaron su relación." ( They finished their relationship ) and it wouldn't be wrong, but it's not that common.
In the case of "acabar", there is one way of using it in Spanish that means something a little bit different, but it's very common, so you should know it, here is the example:
"Acabo de pensar en ella." ( I just thought of her )
"Se me acaba de ocurrir una buena idea. ( I just thought of a good idea )
As you can see, you can use "acabar + de + infinitivo" to express that something has just happened.