I try to learn and understand as many facts as I can. Of course, many people would like to benefit from the opposite, that is, digging into a certain branch as deep as they can.
I try to do the opposite, which I see as my main advantage, as the opposite to professional mathematicians. This is because they have their own careers and has their professional criteria to fulfill (writing articles in journals, gaining citation points, etc.).
As an amateur I am not obliged to do so, and this is a great freedom. If you want to be creative, you may try to dig here and there, and probably you will be lucky to find certain problems which are not penetrated, or you may find just something interesting enough (for example, your own point of view on a well-known area, maybe you find a surprising connection and, even if it is well known, it is funny to discover it once more, etc.) to write it somewhere, maybe on a blog.
In summary: I read as much as I can, I learn as much as I can, and I ask as much as I can.
As regards to low-level entry (you need of course to be genius to discover it, but nothing more;-), an example is Feigenbaum's famous discovery about chaos, etc. As far as I know, he used only a programmable calculator to discover it. He was just inquisitive, nothing more, nothing less.