The plot is a mess, but a happy one; two stories which seem to have very little to do with each other clash.
In the seeming main story, British wizard Newt Scamander (Eddie Redmayne) comes to New York on a mysterious errand. His one suitcase contains a pocket universe filled with magical creatures. Naturally, it gets switched with the seemingly-identical case of one Jacob Kowalski (Dan Fogler), a "No-Maj" - or Muggle - who works in a candy factory, and wants to open his own bakery. After hijinks in a bank, Kowalksi walks off with the magical suitcase, while Newt walks off with Tina Goldstein (Katherine Waterston), an ex-Auror demoted to the wand registration office, who arrests him for not obliviating Kowalski (who has seen Things He Should Not).
Three Fantastic Beasts get out of Newt's case, and the apparent foreplot - which is over about halfway through the movie - is about his, and Kowalski's, attempts to capture them, with lots of physical comedy. Along the way, Tina's sister Queenie (Alison Sudol) is quite taken with Mr. Kowalski.
Meanwhile, one Mary Lou Barebone (Samantha Morton) is preaching in the streets of New York about the evils of witches who Walk Among Us. Her adopted children, including Modesty (Faith Wood-Blagrove), Chastity (Jenn Murray), and Credence (Ezra Miller), distribute her leaflets. She is a cruel "mother" and punishes them regularly, especially Credence.
While she seems at first a nutcase, it turns out that there are things praying on No-Majjes in New York. Mr Graves (Colin Farrell), the head Auror or something, sends Aurors out to destroy whatever-it-is...but he has mysterious connections with Credence Barebone, and seeks someone called "The Child."
Well, it all ties together -- naturally -- and before all is done Newt encounters the President of America's Magical Congress (Carmen Ejogo) and the villainous Grindelwald (the inevitable Johnny Depp).
Production deisgners Stuart Craig and James Hambidge have done a lovely job of giving the movie a look and feel appropriate to the time and place (New York of the 1920s or 30s). There is a nice contrast between a slightly washed-out look to the Muggle/No-Maj world and a very colorful magical world, especially inside Newt's suitcase.
The acting ranges from good to excellent. The standout actors in this production are Fogler and Sudol, who carry an entertaining subplot on their very capable shoulders. Redmayne is, for some reason, made up to look like Matt Smith as the Doctor, which I found distracting at times, but handles his role with some real passion. Colin Farrell is a delight. And Johnny Depp is, fortunately, used minimally, and playing Grindelwald quite straight.
I must give Rowling credit. After the mess that was Harry Potter and the Cursed Child, I had very little expectation of her ability to create a worthwhile new story in the Potterverse. Perhaps because she wrote it herself, unlike Child, Beasts does hold together as a story (or two stories). There might be hope for this trilogy.