Stefan: There are at least two approaches to tsumego.
The first approach is to solve the problem more or less 'on sight'. Speed is a concern - you typically spend 20 seconds or less on the problem (even if that implies not completely reading it out), decide on the right move for the situation, and check your move versus the correct answer. The purpose of this kind of study is to internalise a lot of different situations and get them in your head as working knowledge. A bit like developing reflexes when you learn how to drive a car. I use this kind of study to get to know the basic shapes better and work them faster in games (tripod group, anyone?), as well as to increase my tesuji-spotting capability.
The second approach is to take as much time as you need for a problem, but work it out systematically and don't look at the solution until you've convinced yourself you have completely read out all moves and countermoves (bonus points for working out the value of endgame sequences and how additional moves in the area influence the position). This takes a higher level of concentration. I use this kind of study to increase the accuracy and completeness of my reading (and frankly also to regain a bit the ability to focus, because the fast reflex-stuff makes me a tad sloppy in actual gameplay).
For the first approach I love taking a book a la Graded go problems for beginners and race from cover to cover. For the second approach I like the above mentioned goproblems.com. Solving problems there with a registered account gives you feedback in the form of a rating. And having a rating to maintain or boost tends to cause some of us to concentrate a little better... :-)
Dieter: I think it is very important to continue searching the various plausible moves, even if you have found the solution to a tsumego. This way one learns to know the shape even better. Also, even if "one" move works, there might be a better move. This is one of the themes in the Kanazawa Tesuji Series. I invite everyone to add possible moves that I haven't included and the reason why they fail.
Move order matters. It's much easier to prove that the right solution works against all opposition than it is to find it, so starting out with the right answer saves time. If you don't know where to start, the order suggested above is good (isolate, reduce, then cuts and placements), and straightforward moves before fancy ones is good too. After a failure, try the opponent's successful moves.
The standard advice is to read each sequence once and only once, to completion, and then instantly recognize transpositions when they occur while reading future sequences. I'm not that strong, I find myself backtracking and retracing a lot.
Terminal states matter. Terminal states are where you can stop reading, either because one side got two eyes, ran out of liberties (and no under the stones shape is left), or you recognize the shape and know the result. Static recognition saves work and increases accuracy, and I'm sure stronger players are much better at it than me.
Know your ko assumptions. I.e., first assume that you will lose all kos -- this is optimism, you're assuming that you can live/kill without ko. Therefore, never read down a branch which involves you taking a ko, but the opponent always may. If you win, you have won without ko. If failure, then reverse the assumption, and try to win with ko. If you now succeed, then figure out what kind of ko it is, if a better ko is available, etc. Slavishly following this assumption fails when the final result involves a double ko, but these have to be recognized separately. When thinking about ko like this, local sequences never include ko threats or recaptures, which simplifies things considerably. This might be equivalent to assuming that first your opponent and then you are komaster.
Most problems that I get wrong are due to misevaluating terminal states, usually by missing a shortage of liberties; getting confused visualizing the stones during reading or failing to consider a key move at all is much rarer.
Bill: Let me just add a tip that I read about not too long ago. In a study of eye movements while attempting to solve tsumego problems laid out on a board, there was a significant difference between the eye movements of weaker players and stronger players. The weaker players tended to look at points to play stones. The stronger players tended to look at points to make eyes.
tapir: I doubt that this page is right about professional players approach to solve tsumego. Always reading out every move in every possible branch etc. (this is more like the brute-force computer approach isn't it?) I don't doubt that they read a lot and really fast, though :)
mike?: @tapir, how else can you attempt to read out the possible solutions unless if it's a shape that you recognize and have already read out in order to live/kill. Either way your inference to it being "brute-force" makes it sound difficult, which is incorrect -- it only requires practice to get a better understanding.
tapir: Well, I'm sure professional players read more, more accurate and faster... but I doubt they read all variations from all possible beginnings. Since this is what some beginners try to do... rather they discard more beginnings and variations they recognize shapewise or experience-wise as impossible and read the remaining much smaller tree out faster, more accurate etc. The page about reading contains the same argument btw. If you literally "read out everything" then there is no place for better understanding. Imho better understanding shows in reading the critical spots, exploring the relevant lines vs. reading all possible lines from all (even to the experienced player) obvious failing beginnings.
betterlife: Why on earth should a professional player explore each move with all its branches?
OneWeirdDude: Is there a part about playing tenuki once you realize the group you were trying to save is dead, so as to save a ko threat?