This is a kind of genre thing, meaning you either like the 1950s musicals or you don't. If you do, you'll love this. Personally, I prefer the 1930s and most of the '40s musicals with the dancing talents of Astaire and Rogers, and Eleanor Powell, Bill Robinson, Ruby Keeler, James Cagney, Shirley Temple and so forth but the songs of the '50s, the slower dance numbers and the soapy melodramas of the decade all turn me off.
This film is a case-in-point. The first song was okay but the next three did nothing for me. By then, the story didn't have much appeal, either. The presence of Deborah Kerr is another minus. I don't think I've seen a movie she starred in that I liked, including this one, where the goody two-shoes English teacher she portrays spends half the movie threatening to leave Siam. (I which she had!).
However, divorcing myself from likes-and-dislikes, there is no denying this Rogers and Hammerstein production has a lot of appeal to many folks, particularly those who liked "The Sound Of Music" a decade later. There are similarities in the R&H musicals. Thus, if you liked the Julie Andrews flick, you should like this, too.
This is a Lavish production with, yes, a capital "L." This is the kind of big-production musicals you rarely saw after that generation. You also get the dubbed singers, unlike today, where the actress isn't able to really sing so Marnie Nixon comes to rescue of Kerr, as she did with Natalie Wood in "West Side Story" and Audrey Hepburn in "My Fair Lady."
Yul Brynner is "King Mongkut" and is the stereotypical traditionalist, the kind filmmakers always portray in a negative way. He isn't "progressive," as the left wingers like the say, but the education teacher (Kerr, as "Anna Leonowens") will set him straight. Secular-progressives of today always place teachers higher than people trying to cure cancer! However, Yul is good in this role and even employs some comedy along with his more-bark-than-bite character. Justifiably, he is the big star of this film. Brynner had magnetism. Even in "The Magnificent Seven," Yul was the one cowboy who mesmerized the audience.
In summary, it's a fine movie for its day and millions of people enjoyed it. I'll leave it at that.