I saw a good chunk of this movie on PBS when I was about ~10 years old, but this past week was the first time I actually sat down and watched it beginning to end. The parts I remembered from childhood include: the space station spinning to Strauss like a ballerina, the mysterious black monoliths showing up, and (most vividly seared in my memory), HAL singing "Daisy Bell" as he's being shut down.
I like most of this movie a lot. It achieves an epic feel without resorting to the usual glittery tricks other epic films employ (lush scenery, dazzling costumes, star actors to deliver crackling dialogue). What it lacks in instant gratification, it makes up for in thought provocation. The pacing is solemn and meditative. The starkness of it leaves you feeling unsettled.
I would put this movie in the category of "films you should watch once a decade." It is the work of a supremely confident director (if you ever needed proof that Stanley Kubrick was born to make movies, here it is. But I doubt you need it.) Kubrick was still in his 30s when he started production of this film. Pretty impressive.
Fun fact: the apes in the beginning section were portrayed by a mime troupe.