Learn Spanish (Latin America): Voy a empezar a coleccionar fotos de la ciudad. - I am going to start collecting photos of the city.

In English, we often say "start doing something" using an -ing word. In Spanish, the phrase "to start doing something" is formed with empezar a followed by an infinitive (the dictionary form of the verb, ending in -ar, -er, or -ir). Because of this rule, you cannot use the Spanish -ing form (coleccionando) here.

Why are there two a's in this sentence (Voy a empezar a...)?

You are seeing two different verb phrases stacked together! First, voy a (I am going to) always requires an a to connect to the next verb. Second, empezar a (to start) also requires an a before the action that follows it. Since empezar is in the middle, it gets an a on both sides.

Is it ever okay to drop the second a and just say "empezar coleccionar"?

No, the a is mandatory. While some Spanish verbs connect directly to an infinitive without help (like quiero coleccionar - I want to collect), empezar is a verb that must always use the preposition a to bridge the gap to the next verb.

Could I use comenzar instead of empezar here?