Firstly, just because you have a PhD in some (probably abstract) area of math, doesn't mean you automatically have a gift for everything to do with numbers. So give up on this idea that you're sitting on a money printing machine just waiting to be plugged in. Just like you've undergone 5+ years to do your PhD, on top of 4+ years of build up during undergrad and possibly 12 years during K-12, the "market wizards" have also spent a lot of time learning about finance to get to where they are. Not to mention inborn talent (for abstract math in your case, for finance in theirs).
By your own admission your experience and knowledge of trading is minimal, so get ready to start at the beginning. The good news is, this is where your PhD will afford you some benefits:
- You are fluent in mathematical terminology, accustomed to reading proofs and theorems, and (hopefully) comfortable applying principles given in a very abstract form. This is a huge barrier for non-math people that you won't have to overcome, so try to go for more mathematically rigorous aspects of investing like quant finance, statistical finance and academic papers on pricing/portfolio optimization theory. You will learn these much faster.
- Your research experience should have taught you how to digest technical texts quickly, take good notes, think critically about your text, and easily spot illogical quackery. Use these skills.
- Learn how to code and work with big datasets. Learn at least basic stats that goes with it. Luckily in finance your statistical approach is not bound by the reviewer's whims, if it works, it doesn't matter what some other professor thinks of your particular p-value calculation method. But it still has to work. More broadly, these days you can't do any useful quantitative work in trading without big data (it's all been arbitraged away, and the markets are too complex). So you have to get comfortable working with big datasets, and also obtaining them (often financial data is locked behind paywalls).
- The latest big thing these days is AI, specifically convolutional neural networks, deep learning, machine learning generally and all that. This would be very useful to learn. Your existing math knowledge may or may not carry over much, but regardless you are in a good position in terms of fundamentals to have as good a shot as anyone to learn ML. I recommend you try to learn as much ML as you can, trying to stay focused on time series, and ideally finance/stock data. A lot of ML is pretty general, so if you learn to predict say Taxi commutes or cancer treatments, it's not that different from analyzing stocks, but when getting started it can be very helpful to learn in the appropriate domain and hearing about all the particular quirks of that domain.
As far as monetizing your PhD, there are two ways to go: You can trade yourself, or you can work for a company doing a tiny part of a big trading system very well. I highly recommend against the first option. You would be competing against some real clever dudes, who on top of experience and knowledge have immense institutional resources and connections behind them. You will likely fail. At best you could try a niche, non-traditional market like cryptocurrency, but then you have all the problems that come with that. And the big boys have all heard of crypto now, it's not 2009 anymore.
The more boring but much more promising route is to brush up your skills as much as possible and try to get a job in the industry. Some companies don't want random dudes with a PhD and a head full of dreams, they want people with finance degrees and finance experience - you'd be wasting your time talking to them. Others are willing to take promising people with good skills, but no or little finance background - this is what you're going for. They would be hiring you mainly for those skills (probably coding, data, math, stats) that they're looking for, not finance knowledge, but having finance knowledge would probably make a considerable difference during hiring, so your strategy would be to learn the practical aspects of your existing skills, learn to communicate effectively what those skills are and what your background is (the interviewer probably won't have a clue), but also as much basic finance as you have room for. Working at a company or institution will shield you from market risk, but people working on things that make a lot of money, usually also make good money themselves. You can then save some of that money and do your investment, too.
I know this might sound a tad vague, but writing an entire math to finance transition curriculum here would not be feasible, nor would it be a very helpful answer to other users because it is so dependent on the person. Good luck!