In addition to Acts 6, there's good information on deacons in Phil 1 and 1 Tim 3, which Fr. Enright refers to in the first essay in the Deacon Reader.
Phil 1 specifically addresses the "bishops and deacons" in this city. They are described as having a "partnership in the Gospel" (1:5; and 1:7, "defending the Gospel") and as being completed by Christ's "good work in you" (1:6 - a prayer still used in ordinations: "May he who began the good work in you bring it to completion"), and as "sharing in God's grace with me [Paul]" (1:7). All these ideas resemble the actual ministry of the Seven, and will certainly become characteristic of the ordained by the end of the 1st century. But there's a little element of doubt that "bishops and deacons" are, this early, recognizable offices in the Church, distinct from others, because (a) the Apostles are still alive, and (b) Rom 16 refers to Phoebe as a "deacon". So the words "bishop" and "deacon" seem to be used to refer both to those on whom hands had been laid for the apostolic ministry (ordained, like the Seven), and on those on whom hands had not been laid for the apostolic ministry. In any case, even if, before the 60's, the identity of the clergy was still rudimentary and developing, the special share of the apostolic ministry given to some was notable.
1 Tim 3 also describes "bishops and deacons," given a list of moral and spiritual qualifications. This seems pretty clearly something different from baptism; even this early, it looks like, leaders of the fledgling Christian community were expected to be held to a demandingly higher standard. There is also a distinction, as with the Seven, between the bishops and the deacons, since there are two lists of these qualifications, not just one. The list for deacons (vv. 8-12) is still recognizable to us, and, in its essence, is what we're still expected to represent.
These three critical passages, then, seem to indicate that, even in the first generation of the Church, from the 30's to the 60's, the most basic truths of the sacrament of Holy Orders existed in the Church. Even though everyone in the Church was active in charity and ministry, some people were set aside by the Apostles, by the act of laying on of hands, to exercise a special share in the work Christ gave them specifically. There was the distinction between sacerdotal need (Apostles and bishops, which becomes what we call bishops and priests after the Apostles die) for "breaking bread" (the Mass), and diaconal need for preaching, evangelizing, baptizing, and so forth (deacons, like Stephen and Philip in Acts 7-8, from which all the minor orders, now reduced to lector and acolyte, eventually derived). There were moral and spiritual qualities looked for, which not all the baptized were expected to have. All this is part of the apostolic deposit of faith, but the precision of theological understanding and terminology developed in the next 2 or 3 generations.