It varied. Most of the ideas of modern Christianity follow the line Paul took, but there were others. The Ebionites, for instance were strictly monotheist and non-Trinitarian: they were effectively Jews recognising Jesus as the Messiah. Marcion was at the other end of the spectrum, holding that Jesus was sent by a previously unknown God to correct the errors of the OT god, so introducing an explicitly polytheist version of Christianity. The Gnostics (some of whom formed a syncretic religion with Christianity) held that there was one supreme and perfect god, but also "emanations" from this god, one of which created the creator of the material world. Then according to the author of Acts, even Paul played some games in the area: in Athens he co-opted worship dedicated to "an unknown god" (Acts 17:22-24), although it seems very unlikely that he believed that other gods existed as gods.

I am no historian, but the contained in the Bible is language that suggests that the Jews & later the Christians did not recognize other gods as being real.

Isaiah and Jeremiah talk about the "true God", and "the onw true God" Jeremiah 10:10 Isaiah 65:16

John echoes this : John 17:3 Now this is eternal life: that they know you, the only true God, and Jesus Christ, whom you have sent.

and Paul: 1 Thessalonians 1:9 for they themselves report what kind of reception you gave us. They tell how you turned to God from idols to serve the living and true God,

I can see I somehow ended up late to the party, but the answer, as pointed out by some people, is that it was a bit of both. Up until the fourth century, there were dozens of Christian sects spread throughout the Roman Empire and even into the far near east regions, such as where present day Saudi Arabia was. As a result of this, cultures adapted to the spread of the "gospel" of Christ in a variety of ways. One of those ways, was the blending of common religious beliefs into Christianity which was especially easy since there wasn't an official "canon" as we have it today. Specifically, there were churches near present day Syria who believed that there were't just one God, the father, son and holy spirit (doctrine of the trinity was still a foreign concept to most christians of the early church), but they believed in about 30 different gods, and they also incorporated many others into their belief system of Christianity. The first book I mentioned below goes into detail about this and many other sects.

There are three books worth reading. The first is Bart Ehrman's Lost Christianities: The Battles for Scripture and the Faiths We Never Knew which explains this question in extreme detail.

Another book which emphasizes this answer similary is John Dominic Crossan's book The Birth of Christianity : Discovering What Happened in the Years Immediately After the Execution of Jesus. This doesn't go into as much detail into the various religious beliefs as Ehrman does, but it does discuss it.

And finally I'd recommend Robert L Wilken: The Christians as the Romans Saw Them. I recommend this because this book sort of answers this question, but from the opposite direction.

As a similar question, did ancient Jews (say pretty 100 b.c) believe they were worshipping their God to the exclusion of other, existing Gods?

Yes, Yahweh was the "patron God" of the Hebrews, they worshipped Yahweh but accepted the existence of other gods, a belief system known as henotheism.

The clearest evidence of this is Genesis. The 10 Commandments says thou shall have no other gods BEFORE me, which is to mean that other gods exist, just as lesser to God. Also, think back to the story of Moses... The pharaoh turns the staff into a snake. This was believed early on that the pharaohs god has real power, but God is more powerful. There are other examples in the Old Testament too.

As pointed out by others in this thread, Pauline Christianity wasn't the only Christianity. Even Christianity's earliest writer, Paul himself, mentions that there are already other Christianities by his time. Schisms existed already, but the earliest surviving written works of the other Christians were written later than the canonical works.

The short answer is that it is virtually impossible to know exactly what other first century Christians thought about other gods because we don't actually have anything that they wrote. However, it is safe to assume that the earliest Christians, being first century Jews, probably held the opinion of most first century Jews and did not believe that other gods existed.

As Christianity expanded into Asia Minor, Syria, Greece, Egypt, and Rome among polytheists, we find an increase in polytheistic Christian sects as they reached out into polytheistic areas of the world.

Follow-up question: How about the Norse gods?

So 1667 isn't very close in time to "early" Christians, but in Paradise Lost (published that year, hardly the most authoritative source I know) Milton, a partially trained priest, places very many of the Egyptian, Assyrian, and Babylonian Gods (among others) in pandaemonium with Satan, suggesting that all these other gods occupy some ontologically relevant space subordinate to Satan in the Christian "pantheon."