Series is a collective noun, so it is singular; therefore, you would say, "My favorite TV series is Black Mirror."
Now on to the more complex part of the question...
A TV show is any non-movie program shown (i.e. broadcast) on (via) TV, though there are tv shows that have a host and show different movies each episode. One example of this would be 'Elvira's Movie Macabre', in which Elvira, a vampiress, introduced the movie and told jokes or commentary around commercial breaks while showing a different horror movie each week.
A TV series is any show or programme, usually non-fiction though not exclusively, which runs different episodes on a regular basis. It could be a drama, comedy/sit-com, soap opera, vareity/talent show, reality show, news/satire, etc. It could even be a movie if it were made for TV (for example, the Star Wars franchise wouldn't be a TV series because it was made primarily for movie theatres; however, a mini-series, like 'Lonesome Dove' would be a tv series). Some examples of TV shows would be 'MacGuyver', 'Arrow' or the satire/news/comedy shows 'The Daily Show', or 'Saturday Night Live'.
A Serial is a show which is shown in order, as if each episode were a different chapter of a book. Some examples of a serial would be the drama 'Downtown Abby', the Amazon Prime mini-series 'Jack Ryan', or the reality show 'British Baking Show'.
TV series usually have an overarching story with character profiles and histories, but generally the viewer can watch any show in any order and have a basic understanding of the program; whereas a serial should be watched in order because each episode builds upon the unfolding plot of the previous showings. Unless someone is being specific, like a television critic or a student writing an essay, most people use the less precise 'tv series' or even simply 'tv show' to describe all of the above.
