The things that John is really seeing are remnants of the once-bustling city of New York. We know that John crosses the Hudson to get to a city with tall buildings, and before even seeing the city itself, John tells readers that he left the god-roads because they are not easy to travel on.

. . . we do not use the god-roads now for they are falling apart into great blocks of stone, and the forest is safer going.  

That sentence is a good indicator that the god-roads are paved concrete or asphalt roads. Dirt roads get rutted. They do not fall apart into big blocks. Old concrete does break into large blocks that resemble stone.

As John approaches the actual city, he mentions that he sees god-roads that stretch across the Hudson. He says that they are also broken roads.  

I saw that once there had been god-roads across it, though now they were broken and fallen like broken vines.

God-roads that reach across rivers are bridges. New York has a lot of bridges that access it. Many of them are suspension bridges, which is why I believe that John specifically mentions vines in the above quote. Broken and strewn steel cables could look like broken vines.

As for the tunnels and caves, John is talking about the New York City subway system. One good piece of evidence for this is the "temple" that has all kinds of stars across the ceiling.

A mighty temple it must have been, for the roof was painted like the sky at night with its stars — that much I could see, though the colors were faint and dim. It went down into great caves and tunnels—perhaps they kept their slaves there.

The building that he is referring to is Grand Central Station. Its ceiling is decorated with stars and other celestial items. It descends a long way and leads to a plethora of train and subway tunnels.

When John gets to the Place of the Gods, he sees many of what he calls “god-roads.”  He also, at one point, goes into a building which he thinks is a great temple.  He sees that there is some way in which it “went down into great caves and tunnels.”  The god-roads are really just streets.  They may be elevated roadways, but there is no way to tell.  The caves and tunnels are the places where the railroads were under Grand Central Station.

How do we know this?  We can tell that the god-roads are really just roads because John thinks that the Place of the Gods (which we can tell is New York City) was inhabited by gods.  Therefore, any roads there would be, to him, roads for gods.  It may be that the god-roads are elevated because John reaches one of them after “keeping to the heights of the ruins.”

As for the caves and tunnels, we can know that these are Grand Central Station because the “temple” has a night sky with stars painted on the ceiling.  Grand Central Station is decorated in this way.  If we did not know this, we could still assume that it was a major railroad station because we know that New York has such things and that they would look like caves and tunnels to someone who knew nothing of trains.