Standard answer – the church.
Bigger picture answer – God empowering his people to embody the kingdom of God.
Acts is about the church (the gathered people of God) but that is only one part of a bigger picture. Have you ever noticed how little the book of Acts actually says about worship, assembly, church leadership, etc? It wasn’t intended to be a manual on “doing church.” There is more to it than that.
The first thing we have to realize is that there is complete continuity between the Gospels and Acts. Luke-Acts demonstrates this really well as Luke purposefully makes these connection points between his two books but it is even more connected than Luke-Acts. Acts is the direct continuation of the Gospel story, the story of Jesus inaugurating the kingdom of God on earth as it is in heaven. So the church is a part of the kingdom of God but does not represent the totality of all that the kingdom of God encompasses.
In Acts we see the apostles doing the work Jesus sent them to do and empowered them to do by the Holy Spirit. What results is in part church but, again, it is more than that. What is being demonstrated and carried out in Acts is the working of God’s kingdom in the world to draw people from the world into the kingdom of God. We see that in the teaching, which is more often than not to non-Christians (usually Jews and God-fearers) than it is to Christians of what we have recorded. We see it in the miracles and how they demonstrate the power of God’s kingdom against the principalities and powers of this world (casting out demons, healing the sick, and even raising the dead).
What we see in Acts is a direct confrontation between the kingdom of God and the kingdoms of the world. The church is in the mix but it is about more than just the people of God. Like the Gospels, Acts is also about the kingdom of God and what it looks like when God’s kingdom operates openly and powerfully among the competing power systems and principalities of the pagan world to ultimately bring everything in submission to the true king of kings and Lord of lords, Jesus the Christ.
Last, Acts 1:8 is the outline and direction of Acts but still doesn’t outline the reason and purpose within that one verse. That verse sends the message loud and clear where the Gospel is headed and with it the spread of the kingdom…that the Gospel must spread to the “ends of the earth” (which is why it ends with Paul in Rome). Why is that important? It is important because this is what happens when the kingdom of God/heaven intersect with the earth…the effects of the Gospel continue the spread and we see the growth of the kingdom of God among the people of the earth. This is what the kingdom does. This is what the kingdom looks like. That is good news and God’s people, the church, get a front row seat to watch it all unfold.