"We are going to start eating."
Translation:Nosotros vamos a empezar a comer.
The important thing to remember is that the "a" in "empezar a comer" is attached to "empezar" and not to "comer". (Empezar + a + infinitive). The "de" in "acabar de comer" is attached to "acubar". (Acubar + de + infinitive). "Empezar" always needs the "a" and "acubar" needs "de". It is just a rule of Spanish that certain verbs need the "a" and others need "de" (and some do not need either)
In other activities within this same lesson, phrases such as "Ellos van a terminar de comer" have been translated as They are going to finish eating. Why was that not considered incorrect? Why no need for the "a" between the infinitive "terminar" and the infinitive "comer" in this case? The inconsistent translations are maddening.
This site was pretty helpful for me - it shows a list of verbs and the use of prepositions with each. http://www.cliffsnotes.com/foreign-languages/spanish/spanish-i/prepositions/preposition-use-with-verbs
Voy a and vamos a are idioms. "I am going to ..." "We are going to ...." That is why the first "a' cannot be omitted. #2 The second "a" cannot be omitted because empezar takes the preposition "a" before an infinitive. There are about 20 verbs which follow that rule. My grammar books list them all. In addition, there is another verb group that demand a "de" before an infinitive. Elos empezaron a estudiar anoche. (They began to study last night.) 54 day streak. Looks like you' re hooked.
And lots of people seem to be using a lot of other language sites too. For me just plowing or crawling through all the DL lessons is all I can handle right now, and translations are just so impossibly time-consuming I can't even go there yet. One reason I've been slowing down in the lessons is that I've started writing things down, tons of things, for future reference, and that's been so helpful, but makes it really slow.