A lot of it becomes clearer if you read Arthur Clarke's book that the movie was based on ... and some of the questions (why did HAL kill off the crew) are explained in the follow-up movie "2010" (Directed by Peter Hyams - so it was nowhere NEAR as much of a mindfuck).

...regarding David Bowman's Final Trip ...

The Monolith(s) are (among many other things) intelligence generators.

They appeared next to the tribe of Neanderthals, did some minor modifications (explained in the books), and all of the sudden the monkeys are using tools and beating the shit out of competing tribes to take control of the only water in the area.

The monolith creators buried a domino block deep in the lunar surface, and marked it with a significant magnetic pull ... on the chance that the experiment down planetside worked, the monkeys developed tools, and got into space where they could detect a magnetic anomaly, and then go up and un-bury it. As soon as they did, and sunlight hit it (as shown in the movie) it broadcast an incredibly powerful radio signal aimed at Saturn. (In the first book and first movie it was Saturn ... the second book and movie moved this to Jupiter; probably because the Voyager data indicated that Europa would be a fantastic place for life to develop. Just listen to the Jedi, and accept that it was always Jupiter. These are not the droids you are looking for.)

So ... Bowman goes out in a pod and gets close the Monolith ... then all the pretty colors. According to Clarke -- the Monolith pulled out another of its tricks and opened up a hyperspace gate, that sucked him inside. ("My god, it's full of stars.") He was transported 'elsewhere' ... and when he got there, the Monolith creators built a very fast construct to make Bowman feel comfortable as they studied him. Hence: the hotel room - with imagery drawn from video recordings from just about when Bowman took off on his mission (i.e. things the Monolith could record at the time it was uncovered). Unfortunately - the food had no taste-- it was engineered for Bowman's biology, but they didn't know how to build taste into it. The fluids (wine, water) had zero taste -- distilled water). Bowman knew damned well that an intelligence was at work, and had built the construct for his benefit. Bowman was just waiting for them to show themselves.

When Bowman went to sleep, they appeared ... they downloaded his brain ... they interacted with him in dreams (in the movie - the imagery of him looking at himself getting older and older)

... and after they had the memory dump, they evolved him. In the movie -- the spaceborne baby -- which was effectively what Bowman was. A new life form, just starting out.

The post-Jupiter stuff has triggered endless debate; I won't pretend I understand it. But the greatness of the movie as a whole is its theme: not so much intelligence per se, as humans and the tools they use to live. (Note that the music as the final credits roll is "Blue Danube Waltz" -- suggesting that this relationship is something like a dance.)

The Monoliths were created by an extraterrestrial race that deliberately encourages sentience in promising species in countless planets ("And sometimes, dispassionately, they weeded", according to the book). At least for humans, each time a Monolith appears, it signals a change in the relationship between hominids and tools. The first purpose of the Monolith (in Africa millions of years ago) is to make the proto-hominids able to use tools. Its later appearance signals a new challenge: humans have taken their relationship with tools as far as it can go, and to keep the current relationship is to fall into stagnation and ultimately death. So can they eventually abandon the tools? Is the human greater than the tool?

HAL tried to kill everyone in order to blot out the fact that it made a mistake; it went "insane", because it couldn't deal with that fact. That's about it.

This site offers about as plausible an explanation as any about the final scene's meaning. But this is the long essay that really gave me insight into the film; I can't recommend it enough.

I understood the Monolith on Earth evolving intelligence into the hominids. I didn't get why the aliens left one on the moon and one on Jupiter, but it makes sense that they're signposts.

What was the mistake HAL made? I was watching off and on, and I guess I missed it.

I didn't remember either, but just checked the High Weirdness Project. HAL predicted that the AE-35 unit, which controlled the antenna keeping the ship in contact with Earth, was about to fail. In fact, nothing was wrong with it. HAL responded by insisting that human error was to blame, because "he" refused to believe he could possibly have made a mistake.

...ok ... that's a take I hadn't picked up on. (Of course it's been two decades since I've either seen the movie or read the book).

As I mentioned above ... in a way the lunar monolith was a sign post, and in a way it was a test. "If you know enough to find this, and dig it up, then you get the clue to take you to the next step."

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I don't remember how the first book handled HAL's issues -- but according to the movie 2010, HAL killed off the crew because he was given conflicting orders.

HAL's purpose in life was (paraphrased heavily) to accurately and quickly process data to generate conclusions.

However ... When the lunar monolith was discovered, the US president and the NCA (National Council on Astronautics) classified the incident. The Discovery mission was already being assembled to go to (Jupiter/Saturn), so the mission was modified -- follow the radio signal generated by the lunar monolith, and investigate further. The three mission specialists were told of the monolith's existence and given full mission briefings. To maintain classification, they were trained separately from the flight crew (Bowman and Poole), and put aboard Discovery already in hibernation. The flight crew was kept out of the loop. HAL, however, was told of the existence of the monolith, and the Discovery flight's secret agendas. He was given authority to continue the mission should the human crew become incapacitated or killed. He was ordered by NCA and the President to NOT tell the flight crew about the full parameters of the mission.

HAL was ordered to provide false data to the crew ... which is in direct violation to his basic programming. As the movie version of Dr. Chandra (HAL's creator) put it ... "He was told to lie ... by people who find it very easy to lie." HAL didn't know how to. Once Bowman and Poole started getting really curious ... and started discussing deactivating HAL (which would make it impossible for HAL to carry out the NCA-ordered mission) ... HAL started doing the math.

The post-Jupiter stuff requires LSD to comprehend. I have this on good authority from folks who watched it during its original theatrical release, and stated that the entire last sequence was specifically for those who went to the theater high.