Why radiation is dangerous
Nuclear power plants cannot be turned off if not needed anymore very easily; they need years to dismount. One example is Chernobyl, where specialists are still needed there to continue to make it safer but the after effects of the disaster still linger. The disaster released 100 times more radiation than both atomic bombs dropped on Japan in WW2.
There was a sarcophagus built hastily to encase Chernobyl but it is crumbling despite work to try and strengthen it, there are still fears it could collapse leading to the release of tonnes of radioactive dust. Work to begin on a new £600m ($1.2 billion USD) replacement shelter designed to last 100 years is due to start soon. If it is completed, all well and good but if there is an apocalypse and humans are wiped out, what does it matter if it crumbles? There is no one left around to worry about the effects of the radiation.
Matter of time
One single nuclear power plant can expel radiation in bigger areas than USA or the entire Northern Hemisphere. Now think what can do 500 + of them can do.
Effects of radiation
Everything is effected; the wildlife, people, atmosphere, and environment. Some radioactive substances can still be radioactive after 50 + years.
When exposed to radiation, it effects living creatures down to the cellular level. It damages molecular structures inside the cells, causing them to divide abnormally or stop dividing all together. This leads to cancers, and with cells not reproducing and performing as they should, ultimately death. It can take days to years to die from radiation sickness, depending on the level and duration of exposure. In a post-apocalypse, treatments such a chemotherapy and surgeries will likely be unavailable. If large amounts of radiation are unleashed in a nuclear blast or meltdown, most plants and animals in the surrounding area will perish, as will creatures where radiation clouds are blown to, leaving only certain insects alive in the aftermath.
The effects of radiation on zombies is not fully known. Without respiration and use of a circulatory system or major organs, radiation cannot easily enter an undead body and won't damage it. If radiation is powerful enough, a zombie's sheer presence will get it contaminated. Given time, the virus in the undead brain will likely succumb to cellular mutation and degeneration like living ones. If not killed shortly, the contaminated zombie can move out of the radiation zone and possibly expose other areas that were not affected, spreading it to previously safe areas and potentially killing survivors without needing to bite them.
So the best way to survive? Get yourself into a underground bunker with a very long term supply of water and food, medical supplies, and engineering suplies with the means necessary to get/produce more. Renewable energy is a great way of producing energy for electricity, such as solar or wind energy. So, if you are lucky to survive the intial infection then best prepare yourself for nuclear fallout.