As /u/telkanuru points out, there was considerable patronage going on in the missionary work of early Christianity. This shouldn't surprise us that much since patronage was part-and-parcel of how a lot of things worked in the 1-3rd centuries.

Within the New Testament itself, it seems like there is some combination of work along with patronage. For example at various times Paul works as a tentmaker, but at other times he does not. Within the Jerusalem church there seems some indication of communal collection and distribution of wealth to help out the poor.

However for most of the first couple of centuries there are few dedicated buildings for Christian gathering. /u/Cherubaal suggests they just converted whole 'congregations of a different faith'. The problem with this is that the dominant Greco-Roman religious practices didn't really gather as regular congregations, so there just wasn't this kind of conversion of a congregation as a distinct body.

By the conversion of Constantine you see that Christians do own some buildings, since part of his policy is to restore buildings taken from them. And in the later period (I'm thinking 300 to 600) you do get some conversion of other buildings into church buildings.

Paul was independently wealthy, or at least well-off; his family were rich enough to send him to Jerusalem for his education at a famous rabbi's school, and he supported himself by making tents. But in any case it doesn't take that much money to travel when you do it on the horses of the apostles. As for churches, they didn't build them, they founded them; a church consists of worshippers, not bricks and mortar. For a place to meet they just used whatever was handy, like the house of a rich convert.