Series vs Serial vs TV show
Solution 1:
As a British English speaker, I would call an ongoing drama with a new story about the same characters each week a series, and a self-contained drama in weekly parts, such as a dramatization of a novel, a serial. However, American sources such as IMDb call the latter type a mini-series, because it is in a limited number of episodes.
Solution 2:
After reviewing M-W definitions of series (def. 1b) and serial (entry 2 of 2, def. number 2) it is possible to conclude that:
A series contains the same characters throughout, but each episode is a different story. So, you can start watching series from any episode (Martial Law, NYPD Blue, Friends, Colombo);
A serial contains the same characters throughout again, but you should start watching from the first episode, because each episode is a continuation of the previous one. A serial, thus, should be thought of as a whole and undevidable story.
Solution 3:
Any continuing program (as opposed to movies or specials) on TV is called a "series". This includes game shows, reality shows, sitcoms, dramas, etc.
A "serial" is a type of series, usually comedy or drama, where the episodes generally form part of an ongoing plot arc, rather than being mostly self-contained stories. In some cases the plot arc might only last for a season, in other cases there's a storyline that lasts through the entire run of the series.
The most common form of serial programs are "soap operas". But in recent years it has become more popular to make serialized dramas and comedies. "Breaking Bad" and "Lost" are some of the most well known examples of serialized dramas. On streaming services, serialized shows are popular because viewers can binge-watch them, making them like long movies.