My belief is that the way to start out in a research career is by working primarily on small and relatively tractable problems, where you have a good chance of success. You then continue the process of self-development, gradually working up to more important problems (which also tend to be more difficult, although, as noted above, difficulty is most emphatically not the same as importance). The rare exception is important problems that are also likely to be technically easy; if you're lucky you may find such a problem early in your career, or be handed one. If so, solve it quickly!

Even later on, when you've developed to the point that you can realistically expect to be able to attack important problems, it's still useful to tackle a mixture of more and less important problems. The reason is that tackling smaller problems ensures that you make a reasonable contribution to science, and that you continue to take an active part in the research community.