Should I be concerned about drinking tap water? I grew up in the United States and have been drinking tap water my whole life. More  recently some of my friends have been making me feel a little bit guilty or awkward about the safety of tap water, usually recommending me to use either use a water filter, or to boil my water. I think this is just hyperbole, but after a long time hearing this from different people, I am beginning to feel unsure.

I know some places in the United States and across the world have unsafe drinking water, but how do I know where the water is safe or not? Are there any certain tastes that are considered dangerous? Is it better for me to just start using filtered 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.