Should I be concerned about drinking tap water?
It really depends on the location. Sciencemag.org has an article on this, which is backed up by this paper. To quote  a relevant part from sciencemag:


  Allaire and her colleagues downloaded EPA’s data and looked at the number of health-related water quality violations for 17,900 community water systems in the continental United States over a 34-year period. Some were for elevated lead levels, the problem in Flint, but the data set also included violations for coliform bacteria—a group of microbes that is easy to detect and serves as an indicator of bacterial contamination in general—nitrates, arsenic, and other contaminants. The researchers combined those data with information from the U.S. census such as housing density and average household income, to figure out which communities were most vulnerable.


The following sentence from the significance page from the scientific article's website leads me to the conclusion that you shouldn't be worried per se, but that it is certainly worth looking into the situation in your area:


  Here, we show that health-based drinking water quality violations are widespread, with 9–45 million people possibly affected during each of the past 34 years. While relatively few community water systems (3–10%) incur health-based violations in a given year, improved compliance is needed to ensure safe drinking water nationwide.


If there is something wrong with the water (as you might find after your research) then see what the specific advice is. Some things may go away after boiling, others may not. 

Use this website by the EPA to search for the 'Consumer Confidence Report' on drinking water quality by the EPA.