The trilogy of movies directed by Peter Jackson are as true to the book as is possible in cinema. There are a few scenes and characters left out of the movies (Tom Bombadil, for example), yet the special effects, superb acting, a moving score and tremendous story make these films modern classics.
Peter Jackson actually filmed all three movies at the same time, even though they were released over a three year period. They can be acquired together in a boxed set from Amazon.com.
There have been several other on-screen adaptations of Tolkien's work. All of them have labored mightily to remain true to the original works, of course with varying degrees of success.
The Hobbit was made in 1978. It is an animated version, and though the graphics are not always as fluid as we might like, they are quaint and interesting. The music as well as the voice characterizations featuring such greats as John Huston as Gandalf and Orson Bean as Bilbo make the tale quite gripping. It is worth the time spent watching and is a good introduction to Tolkien for young people, though not too young. I wouldn't recommend the story for children under 7 or 8 because there are some fairly dramatic moments, as any reader of the book will remember.
"The Lord of the Rings", Ralph Bakshi's adaptation of the book, of necessity, cuts out much of the plot even though it runs for 2 and a half hours and covers only half of the trilogy. Even so, the art work at many points is quite amazing. He works in as much detail as possible and even uses interesting techniques to keep the animation moving, even when figures are standing still. He also uses a process called rotoscope that takes live action figures and paints over them. This is not always successful as we sometimes see a jarring difference between the real rotoscoped secondary characters and the fully animated primary ones. Unfortunately, this version of "Lord of the Rings" is not currently available.