personally I just take the adhoc approach
, as I haven't really come to have a fixed, step by step strategy. I just look at the problem, try and understand what it's asking, and execute accordingly.
As for the question you posted, I would divide them up into two groups: blue and not blue (not blue being red and green). The first time you pick, the probability of getting "not blue" is 4/7, and the second time you pick, the probability of getting "not blue" (assuming you got "not blue" the first time) is 3/6. The third time has probability 2/5. Thus, the probability of getting no blues is:
(4/7) x (3/6) x (2/5) = 4/35
I do agree with you that these questions are on the level of what you'll see on the GMAT, and I think that a lot of people are spending way too much time on extremely difficult probability and not enough time on the more fundamental aspects of GMAT math.