The verbs empezar and comenzar are synonyms: they mean the same. The difference is in the usage: we prefer the verb empezar in Spain, leaving comenzar for literary texts. While in Latin America, the verb comenzar can be used in any situation.

The verb iniciar is more formal and is used especially in political language: iniciar negociaciones, una huelga (start negotiations, a strike)…

With the words that mean actions, it is equal to start (poner en marcha), so using empezar is incorrect:

CORRECTO: El gobierno va a iniciar unas medidas = The government will start some measures
INCORRECTO: El gobierno va a *empezar unas medidas  

Unlike empezar and comenzar, this verb describes a new action, not a routine:

CORRECTO: Mi padre empieza el trabajo a las 10 = He starts work at 10
INCORRECTO: Mi padre *inicia el trabajo a las 10

Both empezar and comenzar form constructions with the preposition a and the infinitive, that correspond to the English start + gerund:

Iker Casillas empezó a jugar en el Real Madrid en 1990 = Iker Casillas started playing at Real Madrid in 1990

The construction ponerse a + infinitivo means the same, though with a slight difference: the action occurs suddenly, without thinking much:

La niña se dio un golpe y se puso a llorar = The girl hit herself and began to cry (= started to cry suddenly)

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