Translation:First we will drink coffee, and then we will talk about work.
There are many alternative ways of phrasing this sentence that are not accepted.
The verb "выпьем" is in its 1st person plural form. So, "we" is the only thing that can possibly be the subject in this sentence. And in Russian the obvious things tend to go without saying.
why not "First will drink coffee and then we will talk about work".
It is also possible to omit the subject in the first part of the sentence in English!