If I don't have a good problem to work on, how to I decide what to work on next? Some people have an inherent solid taste or a well-defined agenda, but for people like me who are more attracted to the pleasure of problem-solving than to some larger goal, it is somewhat more haphazard. One way is to smell the wind. Going to a conference, I can notice which talks attract the most attendees, which questions are most vividly discussed in the hallways, and which papers, in that broad fashionable area, are coming up with new problems and partial results. With a bit of experience, that gives plenty of material for promising questions to work on.
How do I figure out what's been done? Thanks to search engines, it is now not challenging to trace back papers, read abstracts and introductions, and quickly figure out the state of the art. It's not particularly difficult, but it's still hard work that takes some incompressible time and requires sustained focus and energy. For people like me, who are seldom focused enough for a long enough interval of time, graduate students are a great resource.
How do I figure out how to attack the problem? The key to make up for my lack of culture is to have friends who know a lot. (I could also try to acquire additional new knowledge, but my brain seems to refuse to retain new information; an inconvenient but not uncommon phenomenon in mid-career researchers.) Then I simply draw on their knowledge. Whenever I have a question in probability, I ask Yuval Peres. If it's in analytical combinatorics, until recently I would have asked Philippe Flajolet. If it's in computational geometry, I send email to David Eppstein, if it's in scheduling, I drop a line to Yossi Azar, if it's related to bin packing, David Johnson is the man, etc. I thus have a short list of "oracles" who each know pretty much everything that's ever been done in their own area of specialty, and who can give me pointers to papers or to techniques that they think might apply to my problem.
What about using previous work and trying out some techniques to actually launch the attack? There is a systematic way to go about it - strike a balance between looking for increasingly harder input examples and looking for increasingly stronger algorithms, favor an axiomatic approach, extract clean mathematical sub-problems from the larger, more challenging question, etc. This rather entertaining exploration can be done jointly with one or several colleagues and students, and the brainstorming aspect is quite fun.
Then comes the hard, grungy work of actually pinning down the proofs of the lemmas, spelling out the calculations, etc. That can be delegated to studious students (because, you know, they need to learn, it's good for them); and it only remains to criticize their attempts (because, you know, they need feedback, it's good for them) before declaring the result proved.
How about writing a paper? Good writing is not easy, and for me it takes more and more time as I have become more and more of a perfectionist. It takes me forever to write a simple proof of a simple lemma, because I try to find just the right wording. Instead, I have resorted to a solution that takes much less effort: a student writes, I criticize what he or she wrote, and we iterate until some deadline happens.
And that is how, with good enough students and a little bit of experience, it is possible to output research, and even sometimes good research, without really doing much.