You should go to the beginning and then move forward one at a time. Here is my overview of all his movies. I've seen them all, and have seen many of them multiple times. I grade out of four stars.
Kubrick considered his first two features, "Fear and Desire" (1953) **½ and "Killer's Kiss" (1955) *** to be warm-up acts. I'd start with those. You see the promise and the growth from one to the next. They each clock in at just over 60 minutes, so you can get both of these in just over 2 hours.
Kubrick considered "The Killing" (1956) ***½ to be his first real feature. It's a heist movie and a very good one.
Paths of Glory (1957) ***½ planted him quite firmly on the map of first-rate directors at the time, and that's where he stayed. This WWI drama starred Kirk Douglas, who 50 years ago said this was going to hold up through time and be viewed as his best film. I think he was right.
Spartacus (1960) *** marks the turning point in Kubrick's career away from tight and economical to huge and epic. He hadn't made a movie longer than 88 minutes until this one, which clocks in at 187 minutes. It's a long movie and it's dated in many ways, but it still works. Douglas is terrific again, but he was right about Paths.
Lolita (1962) **½ is the transgressive drama about a middle-aged man and his relationship with a 15 yr-old.
Dr. Strangelove (1964) **** was his first real hit, and it's some kind of odd masterpiece of black humor and social commentary of the time. George C. Scott and Peter Sellers give inspired performances.
2001: A Space Odyssey (1968) **** is consistently my favorite Kubrick movie. It's that project that you get to do when you've just made a big hit and now you have the freedom to do whatever you want. He made one of the best movies ever. I've seen this about 10-12 times now.
A Clockwork Orange (1971) **½ is not one of my favorites. I feel it's all style over substance. Many of his fans think this is among his best. I'm happy for them. I've seen it 4 times and it just doesn't work for me every time.
Barry Lyndon (1975) *** is also not entirely successful, but after seeing it again in recent years, I think it's good in the overall. It's over 3 hours, just like Spartacus, but feels longer.
The Shining (1980) ***½ is now a kind of classic creeper that gets handed down from one generation to the next. Try to find even a young teenager who hasn't seen it. Jack Nicholson is terrific as usual.
Full Metal Jacket (1987) **½ Among Vietnam movies, I don't think this one holds up. Apocalypse Now and The Deer Hunter were the two big epics, and Platoon and Born On the 4th of July were sandwiched around this one and were both much better. I especially like the first 45 minutes or so, but the second half is what it looks like when a director has lost his way and isn't sure about what he's trying to say.
Eyes Wide Shut (1999) **** is totally captivating from beginning to end for me. His last feature is among my favorites. It was the first of his movies that I ever saw on opening day, and it lived up to its event status that summer. It's one of Tom Cruise's best performances, and Nicole Kidman and Sydney Pollack are terrific. I love it.