I've been learning over the past few years, and can now comfortably get on the air and have a conversation at 18wpm or so. Or I can do a contest-style short exchange at 25 wpm or so. My learning has been slow, because my practice has been intermittent, but it's not a race.
There are several computer aids to help you learn. Koch and Farnsworth are your friends. I found lcwo.net and justlearnmorsecode.com to be very helpful, but there are others which are similar.
You're right, in that it's not something that requires intellectual thought, but lots of repetition and practice is required to make things fluent and automatic. You don't want your brain to have to think about dits and dahs, and count them up to turn them into letters. There's no time for that when the letters are coming fast and furious. You want to hear the letters directly, and eventually, you want to hear words. That sort of automatic recognition of the letters and words only comes with practice.
The most similar thing I've done before is either learning to play a musical instrument or learning touch typing. Or maybe learning a foreign language, but a language has a MUCH larger vocabulary that must be mastered -- Morse code is only 26 letters, ten digits, and a handful of punctuation marks and symbols.
While eevblog is great for general electronics, there are other boards with more ham radio related content, and a bigger population of Morse code learners and users. Qrz.com has a fairly active CW/morse code forum where you can find lots more training hints.
People can and do argue the usefulness of Morse code in this computerized age. It may not be essential, but it is fun. It allows me to communicate over long distances using a very simple, small, inexpensive, low power transmitter than I can build myself. I've found that the people I've met on the air are consistently polite and helpful to newcomers. Maybe there is a rude and snarky Morse code operator out there somewhere, but if so, I've not found him yet.
Have fun learning the code!