To explain, I have a Master's in Engineering from a well known university. We did a wide variety of mathematical topics, vector calc, perturbation methods, numerical methods, linear algebra, discrete maths, etc, etc. Basically a whole load of math and stats classes (econ was part of the course too). Now I work in finance where I've come across stochastic calculus and a lot of computation related stuff.
And yet I often come across wide fields of mathematics that I've never even heard of. Naturally this is to be expected, since I didn't do a math degree. It was a close call after high school what I was going to study. I was quite into math contests, though I figured formal study at uni would be quite different from solving short trick questions.
Is there a consensus as to what the major areas are? Where will I find the most new and interesting maths given my background?
I realise there's a degree of subjctivity involved.
Some major areas in order of increasing relevance to your background, to the best of my knowledge: logic/foundations, abstract algebra, topology, geometry, analysis (and combinations of the previous).
There are also subsets of things you've seen that have likely been glossed over in the context in which you've been taught. For example, in the realm of partial differential equations, you likely have not bothered with the existence/uniqueness theorems, or dealt with the subtle distinction in considering functions as distributions or within the weak topology, as opposed to the usual framework. Within linear algebra, it is likely that you never discussed vector spaces that weren't either $\Bbb R^n$ or $\Bbb C^n$, though many more useful vector spaces exist. Infinite-dimensional vector spaces are mostly the subject of functional analysis, and "vector spaces" over finite-fields or rings tend to fall under the purview of the "algebraic" areas.
In fact, I wouldn't be surprised if you've already dealt with quite a bit of analysis, if only in passing. It plays a key role in the foundations of both numerical methods and perturbation theory. Measure theory, which I would consider a subset of analysis, is particularly important, not least in anything to do with probability.
There's a lot of interesting math out there, it's hard to say what might be the most interesting for you. There are some things you've seen that are deeper than you might imagine, and there are whole worlds you've probably never been introduced to. Personally, I think Lie algebraic structures are pretty neat.