Psycho-Pass is one of the best psychological thriller anime to come out in recent years, so of course, it spawned several sequels and movies. So far, there are a total of three seasons of the anime. As for the movies, there are three of them, too, but two of them are divided into three parts. There's a lot to cover, which can make the task of covering all of it daunting for newcomers.
The sheer amount of content of Psycho-Pass anime material makes one wonder whether the movies are worth watching. Normally, anime movies tied to a series are comprised of non-canon material that doesn't really influence said series or one's experience with it. However, Psycho-Pass is different; the movies do have a direct connection to the anime and vice-versa. As for whether these connections warrant watching them, that will require a closer look at each season and their associated anime.
The first movie of the series follows Season 2 of the anime. For the most part, it can be watched without watching Season 2, as only a few new characters from that season make brief appearances. However, It's necessary to watch Season 1 before this movie.
The first movie follows the story of Inspector Akane Tsunemori as she takes on a mission in the Southeast Asia Union. Her former partner, Enforcer Shinya Kogami, has been seen working with terrorists in the area, and she makes it her duty to uncover the truth.
This movie is important for a couple of reasons. For one thing, it develops the world of Psycho-Pass and gives audiences an idea of what life is like beyond Japan, the series' main setting. On top of this, it shows what Kogami has been up to since the end of Season 2. These things don't necessarily factor into Season 3, but they do serve to make the world of Psycho-Pass feel much bigger.
The second Psycho-Pass movie, Sinners of the System, is an anthology series with events set between the first movie and Season 3. Each segment of the movie focuses on different members of the Criminal Investigation Department's Division 1. Akane may be present in these stories, but she's more of a background character than the protagonist she was in prior installments of the series.
Besides giving the rest of Division 1 more screentime, this movie helps to develop them more and gives the audience a better idea of what each of them is like. Teppei Sugou, for example, has a whole third of this movie devoted to his backstory; prior to this, fans were given little reason to care about him in the series proper. Likewise, this movie gives members of the main cast besides Akane a chance to show what they can do on their own.
The third movie, First Inspector, is almost absolutely necessary to the plot of Season 3. Not only did that season end on a cliffhanger, but the movie picked up its story right where everything was left off. It provides necessary closure for Season 3's story and all the characters involved.
As for the writing and production quality of these three movies, they're about what one might expect after watching the anime. Naturally, Season 1 was lightning in a bottle that later installments of the franchise tried to replicate but couldn't quite get down. With that said, all of the movies are fairly comparable to their respective preceding seasons; their storytelling and animation quality are all decent in their own right, even if they don't quite compare to Season 1. As long as one goes into either later seasons or the movies with these expectations, they should come out of them satisfied.
Those looking to watch Psycho-Pass should definitely consider watching these three movies. They not only provide decent worldbuilding and character development, but they also have story elements that tie them directly to the main series. As far as entertainment value goes, they're certainly worthwhile in that regard, too. For the most part, adding them to one's watchlist isn't necessary, but it is highly recommended.