On the street where you live
In your neighborhood, critters would wander around, looking and wondering.
First the little ones: mice, groundhogs, raccoons, skunks, foxes and beavers. That last one might surprise you, but North America was once rich with beavers.
Bigger animals would come later – deer, coyotes and the occasional bear. Not in the first year, maybe, but eventually.
With no electric lights, the rhythm of the natural world would return. The only light would be from the Sun, the Moon and the stars. The night critters would feel good they got their dark sky back.
Fires would happen frequently. Lightning might strike a tree or a field and set brush on fire, or hit the houses and buildings. Without people to put them out, those fires would keeping going until they burned themselves out.
Around your city
After just one year, the concrete stuff – roads, highways, bridges and buildings – would look about the same.
Come back, say, a decade later, and cracks in them would have appeared, with little plants wiggling up through them. This happens because the Earth is constantly moving. With this motion comes pressure, and with this pressure come cracks. Eventually, the roads would crack so much they would look like broken glass, and even trees would grow through them.
Bridges with metal legs would slowly rust. The beams and bolts that hold the bridges up would rust too. But the big concrete bridges, and the interstate highways, also concrete, would last for centuries.
The dams and levees that people have built on the rivers and streams of the world would erode. Farms would fall back to nature. The plants we eat would begin to disappear. Not much corn or potatoes or tomatoes anymore.
Farm animals would be easy prey for bears, coyotes, wolves and panthers. And pets? The cats would go feral – that is, they would become wild, though many would be preyed upon by larger animals. Most dogs wouldn’t survive, either.
Like ancient Rome
In a thousand years, the world you remember would still be vaguely recognizable. Some things would remain; it would depend on the materials they were made of, the climate they’re in, and just plain luck. An apartment building here, a movie theater there, or a crumbling shopping mall would stand as monuments to a lost civilization. The Roman Empire collapsed more than 1,500 years ago, yet you can see some remnants even today.
If nothing else, humans’ suddenly vanishing from the world would reveal something about the way we treated the Earth. It would also show us that the world we have today can’t survive without us and that we can’t survive if we don’t care for it. To keep it working, civilization – like anything else – requires constant upkeep.