It appears that in part the OP wants to know how to do "original" mathematics. Well, to raise the bar a little, the usual criteria for publication are new, cor

It appears that in part the OP wants to know how to do "original" mathematics. Well, to raise the bar a little, the usual criteria for publication are new, correct, and significant. So, the OP was asking about "new", but likely he will also want to know about "correct" and "significant". Okay. My suggestion is to do applied math. How? There is a famous recipe for rabbit stew that starts out, "First catch a rabbit". Well, for applied math, we could have a recipe that starts "First find an application" or at least find a real problem that needs a solution. There is a broad range of how to use this recipe -- be in an applied department, e.g., business, engineering, agriculture, medicine, and there get the real problem to start. E.g., R. Bellman was in engineering and medicine. Or, maybe better yet, be in some such field in the real world, find your problem, get at least at first-cut solution, then go to graduate school in such an applied department and use your work as your Ph.D. dissertation. I did that. Next, it turns out, if take what looks, first glance, like a serious applied problem, or, really, nearly any fairly new applied problem, and consider this problem fairly carefully, say, try to find the first good or a much better solution, then likely can find a doable math problem to attack. Automatic Presto! If for that serious/new applied problem get a math result that is new and correct, then the math can be viewed as "significant" due partly to its being significant for the applied problem. "Correct"? Sure, as a means of being correct, do the work with theorems and proofs. Compared with what is available in other fields for being correct, the theorems and proofs of math are a great advantage. Next, often you will find that existing math still isn't quite or at all what you really need for the applied problem and, thus, have to create some new math, that, yes, will be guided by the applied problem and get some of the significance of your work from that problem. And at time can discover mathematical questions that, maybe, are not important for the applied problem but are interesting as mathematics or have some promise for other applications; so, starting with the applied problem can provide an injection of secret stimulant that can make clear several new directions to pursue. Also, since nearly everyone in academic STEM fields or nearly everyone in academics has physics envy, in particular, mathematical physics envy, in nearly every field the work that is most respected is that which mathematizes the field. So, do some of that. So, first get an application! For tools, I would suggest a good background in analysis, say, through Royden's Real Analysis and/or the first half of Rudin's Real and Complex Analysis. Also get a good background in optimization. For a desert buffet of math, especially the Hahn-Banach theorem, useful for applications, take a pass through Luenberger, Optimization by Vector Space Techniques. Then, get a good background in stochastic processes. And have some more, really, essentially anything and everything in an ugrad math catalog. You will find that in applied fields such as I mentioned, nearly all the workers struggle terribly with their math background -- they know that they need much more math than they have, and you will have a big advantage. Use it and, thus, do the original math you want. Do this in a department of pure mathematics? Maybe some such department would like to have some applications from outside the department and be interested, but mostly not. But work in other fields, especially some parts of engineering, has long shown some really interesting and valuable questions and results. E.g., the question P versus NP is now taken as quite serious mathematics, but heavily the question started with integer linear programming in operations research. Some of the work in linear programming and special cases of linear programming on networks resulted in some darned interesting questions with some nicely non-obvious, original answers; in principle many of the questions could have been pursued directly from now classic work in linear algebra but were not and apparently because the motivation was not available or some significance was not clear. So, there was interesting work by W. Cunningham, K. Borgward, V. Klee and G. Minty, R. Bland, D. Bertsekas, and more, and some of this work was done outside departments of pure mathematics. Net, for over 50 years there has been a big theme: Not all the interesting, powerful, valuable, important research in mathematics is done in departments of pure mathematics. Questions?