Hi, 

My question is:

"İçecekler satılmaya başlandı" or "Içecekler satılmaya başladı"

Which one is correct in your opinion? I think the second sentence is correct because içecekler is the subject here. If it was "ben içecekleri satmaya başladım" then it would definitely be active (başladım), but when it comes to the first two sentence I can´t be sure which one is better.

Does one of them have to be wrong?

There are two "passives" in Turkish actually, the classical Latin type passive which exists in English also and the impersonal which can be formed out of intransitive verbs and which denotes an action which is performed by some people who we do not want to define now. I suggest the first sentence

                          İçecekler satılmaya başlandı

represents the impersonal, meaning something like ´some people began the action where drinks are sold´ and the second sentence

                          İçecekler satılmaya başladı

is the classical passive sentence, ´drinks began to be sold´.

Is there a third option

                          ?İçecekler satmaya başlandı     ?

this isnt possible but

"içecekler satmaya başladı" might be used. its meaning:
there were drinks on shelves on a shop and no one have bought any of them for a long time. when the customers start to buy that drinks again, the shop owner may use this sentence. "içecekler satılmaya başladı" is the best and clear form for this stuation but both may be heard.