I love reading the book of Acts in the Bible because it relates their travels within a part of the Roman empire. What strikes me is how similar to modern day traveling it seemed to be, just in longer timescales. But the same way we could say today "I'll catch a plane to Madrid and from there I will proceed to Paris", you read there about people taking ships and then traveling on foot to other locations as if it weren't such a huge deal i.e. not as if they were saying "I'm embarking on this long journey to the end of the world, farewell!"
I guess we could say the part of the world covered by the empire was quite "globalized" at that time, in a sense.
Christianity wouldn't have spread nearly as quickly or as far if it hadn't been able to piggyback on the Roman Empire's infrastructure. On the flip side, after Rome's political influence left most of Europe, the parallel church structure was still there to help maintain civilization at least a little bit during the barbarian invasions and general collapse.
[Leaving out Mithraism, which also spread in similar ways, as well as any discussion of what exactly constitutes "barbarian" vs. "civilization" since it gets fuzzier the closer you look.]
The soldiers across the Empire came from all the imaginable parts of it. Going through the museums in Europe near to the former Roman camps, I've seen the statues of Egyptian gods brought or made by Egyptian-born soldiers or artisans, found between the statues of Mithra.
The Pax Romana was probably a big part of that. During that time a person could reasonably expect to travel through most of the Roman Empire without running into anything worse than local thieves and without having to deal with national borders.
Both before and after that time period in Western Europe, in contrast, any long-distance travel was likely to run into wars of expansion or civil wars, local warlords, or so on.
I guess we could say the part of the world covered by the empire was quite "globalized" at that time, in a sense.