I was a CW operator in the military, so after all the training I did to learn the code and all the copying I did throughout my military career, the code kind of stuck with me and is like a second language. I learned Morse code at the US Army Intelligence School at Ft. Devens, MA. We leaned the code by listening to it through headphones and copying it down on a keyboard. We had to learn it at 22 WPM to pass the school. We did this by copying five-letter groups of random characters, starting at 6 WPM and working our way up. To pass a speed, we had to copy blocks of 300 characters with 30 errors or less, and do that twice in a one-hour period. Needless to say, copying CW for six hours a day, five days a week for six months straight is enough to drive anyone batty, and at the time I never thought I'd ever actually enjoy Morse code. In fact, many of us started calling the army post "Ft. Devenworth." An interesting side effect of the CW training occurred, however. Learning Morse code actually taught me how to type. I was only a marginal typist who had to look at the keyboard until I started the CW school. Learning CW by associating each character with a keystroke taught me how to touch type.
Later, once I had finished the CW school and was working as a Morse Systems Operator (Air Force MOS 207X1), I thought it would be fun to get a ham license and chat with other people using CW. It wasn't until later that I seriously considered it and started studying. The CW part of the test was easy for me. I skipped the 5 & 13 WPM tests and went straight for the 20. The only trouble I had was writing the characters down fast enough without getting cramps in my hand. It's far easier for me to copy it on a keyboard.
It wasn't until I got out of the military that I got my first HF rig and started getting on the air. I had a 2m/44 HT that I had made countless contacts on, but I was finally on HF. I bought a straight key and mounted it on a board that I had beveled, then sanded down & varnished. I operated CW night & day and only occasionally used SSB. I realized that my sending speed was limited by the straight key so I bought a Ramsey keyer kit and set of Bencher paddles. I put the keyer together & hooked the Benchers to it, and was pleased that I was able to easily send at 20 WPM or more. I was limited to about 16 WPM with the straight key. I had a pretty good fist but the keyer spoiled me. When I tried the straight key not too long ago, I had a hard time trying to make decent characters.
Later I got interested in QRP operation and wanted to build a QRP transceiver. Knowing CW came in handy for QRP, since it's a good low-power mode. I have since built several QRP CW transceivers and continuously enjoy using them.
I would encourage all hams to at least try CW. The FCC has finally dropped the CW requirement for amateur HF privileges, but it is a fun mode that is part of our amateur history. And just think, you'll be able to communicate in a way that few people can.
So how does one go about learning CW? There are probably as many different answers to that question as there are CW operators. The most important thing I will say, though, is never, ever try to learn Morse code by dots & dashes. You will never be able to build up speed if you have to think about what a character looks like after you hear it. Instead, learn it by sound. "A" for example should be thought of as "didah," not "dot dash." To this day I have trouble trying to decipher Morse code written as dots & dashes, and that's fine. What matters is that I can copy CW by ear. Also on the subject of habits to avoid is using "soundalikes" to remember what each character is. For example, thinking of "Charlie's got it" when hearing dahdidahdit, or C. Some of the instructors at Ft. Devens gave us examples like that, but we weren't taught them as part of the course. I've heard from other hams who learned CW by memorizing some of those sayings and later had a hard time trying to overcome it. It just adds another mental step which will slow you down as you try to improve your CW speed.
The Koch method of learning a few characters at a time is a good one. I learned this way, starting with a few of the shorter characters and becoming familiar with them. I don't remember exactly which ones they were, but I think they were A, E, I, M, N, and T. These are the shortest characters and would be a good starting point. Once you become familiar with these, then add a few more. Eventually, you'll have added more & more characters until you have learned all 26 letters, 10 numbers, some punctuation, and a few prosigns such as AR, BT, and SK.
The Farnsworth method of learning CW is a system of learning the character by hearing it sent fast, say at about 20 WPM, but leaving long spaces between characters. The idea is to get used to hearing what the character sounds like at high speed, but leaving enough time between characters to slow the speed down. At 5 WPM, there will be long gaps between characters. As the speed increases, the gaps decrease until you are copying CW with normal spaces between characters at 20 WPM (or whatever speed you choose). The main advantage to Farnsworth spacing is that it prevents one from counting dits & dahs at lower speeds and makes you used to listening for the rhythym of the code. Counting dits & dahs is another bad habit that will prevent you from building up speed, and may be the main reason so many CW operators have a hard time trying to get over the 13 WPM hump. It's hard to count dits & dahs at that speed, but if you learn to listen to what each character sounds like you won't fall into this trap.
Quite a few hams don't like or recommend using the Farnsworth method. However, I learned CW that way at Ft. Devens, and it allowed me to build up my speed to 20 WPM before I left the school. I understand its limitations, though. Farnsworth spacing does not teach you good keying habits and proper character spacing. When I'm sending slow with a straight key, I often unconciously send Farnsworth spacing. I will say, though, that if you're planning on becoming a high-speed CW operator Farnsworth is a good choice.
One final thing that all newcomers to Morse code should realize is that learning the code will take time and practice. You shouldn't expect to be able to copy 15 WPM after a week. It will take months or even years of practice & dedication to build up your speed. Learning the characters is easy. Building up speed is hard. It took me six months to get up to 20 WPM, and that was copying for six hours a day. However, I don't recommend anyone practicing that much. The dits & dahs start rattling around in your head after a while, and you'll even start dreaming about Morse code. I recommend practicing for about half an hour each day.
There are many, many Morse training aids available to help learn CW. The ARRL has practice tapes, W1AW sends bulletins for code practice daily, and there are numerous computer programs on the internet for practicing CW with. Most of them are freeware and can be easily downloaded & set up. One of these that I have been playing with lately is G4FON's Koch method CW trainer which can be downloaded at http://www.g4fon.net. This is a fun little program that generates CW randomly or by sending a text file. You can practice using the Koch method in that you can choose how many characters and which ones to practice. If you're having trouble with the characters S & H, for example, you could choose to practice just those two characters. What makes the program fun is that you can choose to add noise in the form of QRM or QRN, make the signal fade, or make it sound like it's hand-sent with a straight key or chirpy transmitter. This simulates real on-air code practice which will make you a better operator. The straight key function is highly recommended for practice in that it will get you used to copying less-than-perfect code.