Chernobyl had a lot more radioactive material than the bombs used again Japan. In addition, the 500-yr number only relates to part of the area; some of the contaminated areas seem to be recovering faster than expected, though some are recovering more slowly.
To go back to the OP’s question more directly: the radiation (long-term) is entirely due to radio-isotopes produced in the blast, which means everything started life as gas that condensed out and mostly became dust that you’d expect to mix with non-radioactive dust, ash, etc. To a large extent, you can follow it just by following wind and water.
Hot spots are definitely the biggest concern. A low point in storm drains would probably have a layer of runoff that is much higher in radioactivity than the streets or surface dirt. In fact, if this radioactive layer is covered up by later layers of mud from runoff, the dirt itself might protect a traveler from radiation as long as he doesn’t start digging around. Of course, post-apocalypse, digging around in cities might have its appeals.
The same issue might come up for enclosed spaces - perhaps a building’s basement gets a full dose of radioactive dust before the building collapses on top. Now the dust has nowhere to go until some luckless explorer opens it up and starts stirring dust around.
Just being exposed to the dust is bad, but you really want to avoid getting the dust inside of you by inhalation. If you’re wearing a breathing mask and carefully wash yourself off after exposure, the radiation you pick up might be fairly limited - just a few hours of time. If radioactive dust lodges in your lungs, it now has the rest of your life to irradiate you. Furthermore, some types of radiation are better than others at penetrating your skin; something lodged in your lung denies even that little bit of protection.
Either way, I think games like Fallout and Gamma World oversell the risk of sudden onset radiation poisoning and understate the long-term effects. It’s likely that exploring an irradiated city would have no effects until you develop cancer 5, 10 or 20 years later.
In terms of acute, sudden symptoms, I think the biggest risk would be from two sources. First, civilian radioactive waste. It might be easy to avoid nuclear power plants and uranium mining and enrichment sites, but hospitals use a lot of radioactive materials. Second would be unexploded munitions - a whole chunk of uranium or plutonium could kill you very quickly. If 20-40,000 warheads are used in an all-out exchange, I think it’s safe to expect a few hundred will fail to detonate and that their bomb casings will be damaged to some extent.