First, while you may be able to find Postgres specific Database Engineer positions - you're probably going to have to switch to one of the big two (Oracle or SQL Server) to have your pick of the job market.
From there, study Relational Database Theory and Database Design (designing databases from the ground up). The book I suggest for beginners is Database Design for Mere Mortals.
You can then branch off into such things as Report Development with SSRS, Crystal, Actuate, etc. or ETL Development is also a possibility. Or you may go the more general route of wielding a heavy and expert knowledge of T-SQL (Microsoft) or PL/SQL (Oracle).
There are many disciplines within the area of databases. Lots of cross training usually exists between these areas though. So every database-related job is different. Some are just focused on a specific area and some require knowledge of everything.
If you can expense learning materials at your organization, I highly recommend Database Design for Mere Mortals: A Hands-On Guide to Relational Database Design. There is also a new edition coming out in December.
As you figured out, this isn't an SQL issue, it's a database normalization issue.
You wouldn't have a table of winners and a table of losers -- you'd have a table of teams, and a table of matches that links them together, and a table of players, and a table of stats. and you'd link them all together using various linking tables.
I used this book when learning about setting up databases.
What you are describing is indeed called data modeling. It is also sometimes referred to as Entity Relationship (ER) modeling. The Stanford course linked to by msjgriffiths below would be a good place to learn, as would Database Design for Mere Mortals. There are lots of books on the subject.
I found this book really useful: Database Design for Mere Mortals: A Hands-On Guide to Relational Database Design https://www.amazon.com/dp/0321884493/ref=cm_sw_r_cp_api_-XVEyb2Y4PZX3
Walks you through the design process, and it's tool/language agnostic, so it explains the concepts without getting into the weeds with the code
Sorry I'm coming in here so late, but "Database Design for Mere Mortals" is where I learned from.
Good tables, good relationships, good life.
Warning: Databases are not easy. There's a reason that some folks specialize in them.
If you're having issues primarily with normal forms and other database design-related topics, I can't recommend Database Design for Mere Mortals (http://www.amazon.com/Database-Design-Mere-Mortals-Relational/dp/0321884493) enough. It is probably my single favorite piece of work on the topic.
That being said, my experience in databases allowed me to not take this course. So, your mileage may vary.
Architecture of a Database System http://db.cs.berkeley.edu/papers/fntdb07-architecture.pdf
Readings in Database Systems http://www.redbook.io
Computer Science 186, 001 - Spring 2015 UCBerkeley Introduction to Database Systems - Joseph Hellerstein https://archive.org/details/UCBerkeley_Course_Computer_Science_186
Stanford's Databases MOOC https://cs.stanford.edu/people/widom/DB-mooc.html
Database Design by Caleb Curry https://www.youtube.com/playlist?list=PL_c9BZzLwBRK0Pc28IdvPQizD2mJlgoID
Database Design for Mere Mortals: A Hands-On Guide to Relational Database Design (3rd Edition) https://www.amazon.com/Database-Design-Mere-Mortals-Hands/dp/0321884493
More at http://Learn.SharjeelSayed.com
I have a similar story, except coming from Marketing and my CIO/mentor IS a BI guy. I've been doing this for 9 months now and can now write entire ETL programs using Python, stored procedures, some DBA, Sql Server data modeling, SQL queries, report building etc etc. And I didn't pay for any training (yet).
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https://www.codecademy.com/learn/learn-sql
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As you probably know coming from the finance dept, change management and user training/acceptance will be your biggest obstacle and most likely YOUR biggest responsibility/role. Really think about how BI fits into your entire eco-system. I just worked with finance (and still am) on choosing an ERP vendor to replace Quickbooks, and we have a large revenue retail company. We're implementing this year.
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Your greatest strength is your knowledge of the business and your end user advocacy. Get some skills to be able to understand the back end and the reasoning, but don't pidgeon-hole yourself into just a technical role. Feel free to PM me if you have any questions! I feel for your lateral career move and it can definitely be a painful adjustment.
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I would highly suggest keeping the composite key for Team_Players (player_id, team_id) as it provides Referential Intergrity and ensures that a player cannot be a part of more than one team at once.
Instead of navigating all of this on your own - I'd also like to suggest picking up a book on Database Design.
Database Design for Mere Mortals is my recommendation. It's a short read, easily digestible and will do you wonders when working to create this database.
If you want to learn it properly (im not sure what you mean by quickly) those 2 books seem good:
https://www.amazon.com/Database-Design-Mere-Mortals-Hands/dp/0321884493https://www.amazon.com/SQL-Queries-Mere-Mortals-Hands/dp/0134858336
On the other hand they are almost 2000 pages combined... So maybe this one might me more digestable in a short time:
https://www.amazon.com/SQL-Minutes-Sams-Teach-Yourself/dp/0672336073
Database Design for Mere Mortals is a good intro to relational database generalities, and Fundamentals of Database Systems is a good one for getting into the internals of them i.e. relational algebra and set theory.
Database design for mere mortals is an excellent reference I've kept throughout the years. It has my recommendation.