First things first, finance is only sort of economics, it's really just finance. I'd highly recommend taking an accounting class (or book) and a grab an intro finance book. Accounting will really help with jargon, and just some really basic things (like balance sheets). Also, "Security Analysis"  is the "only" book you'll ever need, Warren Buffet recommended it to Bill Gates, and now Bill Gates recommends it to everyone.
Back to Economics... There are two primary "groups" of thought... sort of like twins separated at birth who grow to hate each other.
The First: Neoclassical Economics
Focuses primarily on microeconomics and largely mathematical. It's birth is largely due to Economists wanting to make econ a "true science" like we see the physical sciences (biology, chemistry, physics). It starts around the late 1800s and really picks up steam around the time of Einstein. Math was hot and being applied everywhere.
A really interesting period to research and study is right after black Tuesday (and before the great depression) and what the central bank didn't do (before central bank intervention in markets). While I really detest the bastard, Milton Friedman's work on monetary policy is pretty science and generally good here. ,.
I'm a Keynesian (I suppose-- Econ gets deep fast), and so you'd be no where without reading some of what Keynes did to get our assess out of the great depression (i.e. government spending). It's also more or less the birth of Macroeconomics... You'll know you're good when you laugh at forgetting: Y = C + I + G + (X - M). Some good things to get started are looking at the IS-LM  model and AS-AD  model.
That gets you into the 60s - 70s. Tall Paul Volker is the unsung hero of the 80s, read about him (he ran the federal reserve). After that microeconomics starts to fragment into things involving game theory and behavioral economics (Daniel Kahneman is the man).
Econometric analysis mathematically speaking is just multivariate regression analysis for time series or cross-sectional data. More "modern" analysis is probably using panel data  (combination of cross sectional and time series). Calculus, linear algebra, and differential equations should prepare one plenty for everything but panel data analysis. The real "econ" part is applying solid econ theory to the mathematics you're using, a textbook will help . For finance this is your bread and butter.
Game theory will apply a lot of different mathematical tools. You will need to love pure math. To really get into it requires pain or love. I like a healthy amount of both.
The Second: Heterodox Economics
So as it turns out, neoclassical economics is at most half of Economics. It's really where the "philosophy" comes into play. You're gonna need a quick history lesson to sort of see it's topic matter. Economics really didn't exist before... the 1500s. You can try to apply economics to earlier times but you could also just make shit up and post it to twitter. Both would be equally likely to contain truth.
Economics came into existence around the time the Dutch began developing trade routes (1550s). A by product of all this trade, is tons of cash, and goods-- currency (silver, metals, whatever) starts to actually be used in society (before that it was mostly just a status symbol). It pisses off a lot of _institutions_, most of all "the church" and monarchies because money is allowing people to gain power. It's usurping power from them. This is the rise of the "merchant class" and now thanks to money (trade really, but whatever it's complicated)-- people are liberating themselves from the social status they're born into. Eventually modern republics appear, and governments form. Nations trading globally becomes more common (Dutch, English, Spanish) and we get to Adam Smith, David Ricardo , et. al.
Now it's the 1800s. People are seeing the birth and growth of capitalism, industry, corporations, and the tumultuous death of agrarian life. Now the way the "common person" lives their day dramatically changing, for a few it was better for most it was worse. Some economists begin to ask why are we replacing these now defunct _institutions_ with equally shitty, or possibly shittier, ones. This is more or less becomes the birth of heterodox economics which largely studies the more abstract ideas like "institutions"; by it's very nature the content tends to be philosophical.
By the 1920s heterodox economics is falling by the wayside. The content is less able to be tested like a physical science (i.e. no math/stats); so, it's treated like a misbegotten child... By the 1950s heterodox content was marginal at best-- the cold war and fear of communism made (makes) people insane. Economists pretty much had to be pro-capitalism or face being called "commies" and thrown in jail or worse being a narc in a witch hunt. This was more or less the nail in the coffin in mainstream heterodox economics (at least for research in the Occident). After the cold-war ended the nail got pulled out, but I wouldn't say it's really outta the coffin yet.
This book  isn't great but it's quickly digestible and will point you in the appropriate directions.
Some Rambling to Finish
I'd highly recommend not just learning how to use the tools, but why we have them and where they came from. Economics is vastly deeper than the average person will ever know. That depth is greatly empowering and guiding when using its lenses to see and solve problems. One last thing, know there's no going back, you will see the world differently.
 "The Role of Monetary Policy." American Economic Review, Vol. 58, No. 1 (Mar., 1968), pp. 1–17 JSTOR presidential address to American Economics Association
 "Inflation and Unemployment: Nobel lecture", 1977, Journal of Political Economy. Vol. 85, pp. 451–72. JSTOR
 The course I took on panel data, http://web.pdx.edu/%7Ecrkl/ec510/ec510-PD.htm
 He more or less invented trade theory (competitive advantage) https://en.wikipedia.org/wiki/David_Ricardo
Edit: for formatting.
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