Unfortunately, after over 100 years of industrial use, parts of Tacoma have been poisoned by industrial chemicals, heavy metals, and various other toxic substances.
Cleaning things up has happened in places, but in other places the industrial fallout has happened over such a wide range that a complete cleanup is simply not possible. The American Smelting and Refining Company (ASARCO) had a 571 foot (174 meter) tall chimney* that spread the toxic material over a vast area of Puget Sound and Tacoma, and no doubt many of the other companies had similar problems.
In places, particularly public parks, there are signs warning people to wash themselves thoroughly afterwards if they come into contact with the soil or sand inside the park - in order to clean off the toxic materials.
Most likely the same holds true for many areas of Tacoma, such as private residential land near these sites, plus anywhere else that you might come into contact with the dirt or sand.
* When built in 1917, this was the tallest chimney in the world. According to the on-line photo records of the Tacoma Public Library, this was shortened in 1937. It was demolished in 1993.
After living in the tacoma area all my life, (I live in Lakewood, south of tacoma) I should share some of the "danger zones"-
South Tacoma Way- This street stretches from the Downtown waterfront all the way to the Lakewood Industrial area. The worst parts of it are from the underpass on the way to the mall to around where the majority of the auto dealers are. This place is pretty safe by day, but would not go there at night alone.
Tacoma Mall area- This is the surrounding area of the mall. It is between South tacoma and Interstate 5. Parking your car, especially with valuables, in low visibility spots is not reccomended. I suggest that you park near large buildings, such as the Mall or Costco, and only walk when necessary.
Downtown- The hotspot for crime, especially when you get away from the water and Old Town. dont leave valuables in cars here, either.
Wright Park- The major park in north tacoma, between Stadium High and Hilltop. NEVER go here at night. VERY dangerous.
Hilltop, North 6th ave, surrounding area- Mostly residential. There are some wherehouses, and many stores on 6th ave. the part that stretches from "Its Greek to me" to Union ave has some crime, mostly at night. As you get farther from the Hilltop area, it becomes safer.
Generally, follow your instinct, dont go places that seem unsafe, lock your car and keep valuables with you, but other than these places, Tacoma is pretty safe. UP and Lakewood are as well.