As can be seen by the number of answers here, there is no "right" answer to this question. From the beginnings of film, inventors tried to make "color" film using various techniques. Let's talk about those techniques, because choosing a technique determines the film selection. For purposes of this discussion, we'll ignore hand painting and focus on actual color film processes.
The first "viable" color film process was the three-color Lee-Yurner color process, developed by Edward Raymond Turner (cited by flopsymopsy above) with financial backing from Frederick Lee and Charles Urban. However, the "movies" (10-minute segments) made using it in 1902 couldn't be successfully displayed at the time With modern technology, we can sync the three different color images and eliminate the "flicker" that made the film unshakable at the time. Because these movies were not exhibited, they do not have names.
Turner might have fixed this, but he died in 1903. Urban then hired George Albert Smith to further develop the Lee-Turner process. Instead, Smith junked it and developed the two-color Kinemacolor, the first successful color motion picture process, for Urban. As discussed above, the first Kinemacolor movie was the 8-minute short "A Visit to the Seaside" from 1908, which was also the first publicly exhibited film prepared in a color process. Similarly, the first feature-length film shot in Kinemacolor was 1914's 150-minute long documentary "With Our King and Queen through India", also cited above. The problem was, the expense of this project and its public execution, combined with the start of WWI, bankrupted Kinemacolor.
All other color systems from this time were also two-color systems that may have infringed in Kinemacolor's patents (not that they were around to defend them any longer), including the original Technicolor Process 1, developed in the US, which was used to film "The Gulf Between" in 1917. But all two-color systems of this type have a restricted color palate. So Technicolor developed its Process 2, which was also a two-color process, but one which subtracted color instead of adding it like the prior processes. The first general-release film in Technicolor Process 2 was 1922's "The Toll of the Sea". Technicolor Process 3 was merely an advancement of Process 2 (using dye imbibition), so it really wasn't a different method, but it was easier to exhibit and led to huge popularity for color films, as well as the use of Technicolor sequences in otherwise black-and-white films. However, the added cost of filming in color led to Technicolor Process 3's popularity declining during the 1930s, and it still had never been profitable.
In response, Technicolor unveiled its first "true" color system, Technicolor Process 4, in 1932. The first adopter was Walt Disney, who negotiated a 3-year exclusive license for it and used it in his animated "Silly Symphonies" short films beginning with "Flowers and Trees" (1932). The first live-action short filmed in Technicolor Process 4 was "La Cucaracha" (1934), made by Pioneer Pictures (owned by the Whitney family, who acquired Technicolor in 1933 because they loved Process 4), and the first full-length film made in Technicolor Process 4 was "Becky Sharp" (1935), also made by Pioneer Pictures. The Whitneys were also investors in Selznick International Studios, and Pioneer was merged into Selznick International in 1936 -- and Pioneer and Selznick made several Technicolor Process 4 pictures during the late 1930s. ... which is how Victor Fleming, who was directing 1939's "Gone With The Wind" in Technicolor Process 4 for Selznick, came to use the same process in his other 1939 picture, "The Wizard of Oz" (it's worth noting that Fleming also died in 1939, perhaps from the stress of the two movies and the color filming process). Needless to say, Process 4 was profitable despite its high filming cost and brought color movies into the mainstream.
So the answer to the question of the "first color movie" is most definitely NOT "The Wizard of Oz" but actually depends on several preliminary questions: Do shorts count? Did the movie have to be publicly exhibited? And, critically, how good did the color process used for the movie have to be?
If you want to research this yourself, most of the links listed in the posts above are good, but here are Wiki links to the film companies:
https://en.wikipedia.org/wiki/Edward_Raymond_Turner
https://en.wikipedia.org/wiki/Kinemacolor
https://en.wikipedia.org/wiki/Technicolor
https://en.wikipedia.org/wiki/Pioneer_Pictures
