Most shower filters remove chlorine from water. Chlorine is the number one contaminant coming through your shower head. Over time, if enough chlorine comes in contact with your skin and eyes, it can do serious harm. Fortunately, a good shower filter solves this problem.
Arsenic is also in water. You can test the levels of arsenic in your water. Most homes in the U.S. have detectable levels of arsenic in their water. But arsenic, in order to do harm, must be ingested in high levels as a vapor. This can happen over time. It is important to pay attention to arsenic levels in your water. But it shouldn't cause you too much worry.
One way to nip these issues in the bud is to invest in a shower filter that removes chlorine, and be vigilant on your water's arsenic level. The other way is to gain as much information as you can. The NRDC has an insightful question and answer portion of their website dedicated to arsenic in water, and what you can do to avoid it.
The EPA made an informative video that discusses arsenic, its causes, how to prevent it from seeping into your water, and what to do if your levels are high.