Tl;dr : Skip to the last section!

I was good in maths at school, and was silently even proud of it. I could solve problems easily, and would always score well in exams. But it took me less than one exam to shatter the castle of confidence I had built in the air. But just like it happens in other cases, destruction brings along revelation, which in turn brings development!

It was not many years ago, when I decided to appear for an entrance exam for Masters at IISc. The subject was Mathematics, which was the best bet given other options (Physics, Chemistry, Biology, etc.). Little did I know, this exam was going to change my life. But the change was nearer; it wasn't the question paper that shook me, but the first two pages of a book I was reading while preparing for the exam. This was the first of many books listed under "Suggested Reading" on the exam's webpage. The first page was very interesting. I had never before in my entire life faced such beautiful maths!

The second page was a killer to me. It had a proof to "the square root of 2 isn't a rational number". I had always s̶t̶u̶d̶i̶e̶d̶ l̶e̶a̶r̶n̶t̶ k̶n̶o̶w̶n̶ crammed about rational and irrational numbers, I even knew that fact about square root of 2, but never before did anyone teach me how to even ask for a proof, let alone prove it!

It was simply elegant. It made me realize how weak my fundamentals were. In fact, there was no foundation at all. I quickly went through the rest of the book and it took me no time in realizing that all the mathematics I've ever learnt was a sham! That book wasn't about high-level mathematics, but the fundamentals -- about sets, about least upper bounds, about the annoying commutative and associative laws that I had always hated for their uselessness. Suddenly, the revelation bell rang, the castle collapsed, and my feet touched ground.

This was a life-changer, it made me rethink about what I know, and made me crave for the truth, for what needs to be unlearned and relearned. I was prepared to unlearn, and to relearn. It was then that a strong urge to learn basic, real mathematics came to me, and I sincerely hoped for someone to teach it to me someday. About the entrance exam -- I failed it! But thankfully, I had in parallel taken the GATE exam, which qualified me for the interview at the department of Computer Science and Automation at IISc. I was now a masters student there.

IISc is a wonderful place for mathematics. The professors here pay heed to strong foundations and encourage students to take courses in maths. I took one course in the department of Mathematics, and guess what it was about? Real Analysis, taught from the same book which once woke me up. I was now going to learn that subject from a real mathematician.

I started liking mathematics, and over the years, developed some strong fundamentals about the subject. There were many game-changers after Real Analysis -- Probability Theory (the measure theory part took me 1.5 years to understand), Discrete Structures, Mathematical Logic, Optimization, etc. I am still a novice student of mathematics and have much more to learn than I could imagine. But I can advise you on some points, only with the following words of caution: I may be wrong! So, follow them on your own risk.

1. Do not let people convince you that pure mathematics is for mathematicians and scientists and not for engineers. The first step to approach mathematics is by liking it. And you can like someone only if you "know" them, not "use" them, right?
2. Divide your interaction with mathematics into two groups: casual interaction and formal interaction. The former will bring you closer towards maths; you will like it more and more. And the latter will make you stronger and improve your skills: thinking, proof techniques, applicability, etc.
3. Casual interaction: Read some fun books, watch documentaries, movies, series, etc., that would develop your liking towards mathematics more and more. This will not improve your mathematical skills, but will be fun, and will give you more reasons to pursue maths! Some of them I've mentioned in point 5 below.
4. Formal interaction: Pick up some basic books and start studying maths (I've listed some basic subjects and books in point 6 below). And this is very important: write, write as much as you can. The more you write mathematics, the more you become skillful in it. Very few books give elaborated proofs or definitions; so it makes a great difference when you write and unwind things. It makes them simpler to understand. And you will have to compromise on your comforts. You may have difficulties using Greek notation, but let yourself loose and write them. Practise them. And take pleasure while doing so.
   
   Apart from that, also watch some video lectures, but not without a pen(cil) and a paper to take notes. Have this habit of taking notes, even when you know you'd throw the paper away later. Better is to make a soft copy of the notes later, for your own sake initially, but you could share it with others later. LaTeX is one useful tool you'd love while you make notes.
5. Fun books:
   
   Logicomix - An Epic Search For Truth by Apostolos Doxiadis and Christos H. Papadimitriou, et al.
   

1. ​
   
   How Mathematicians Think - Using Ambiguity, Contradiction, and Paradox to Create Mathematics by William Byers
   

1. ​
   
   One, Two, Three, ..., Infinity - Facts and Speculations of Science by George Gamow
   

1. ​
   
   
   Documentaries:
   
   The Story of Maths by BBC
   

1. ​
   
   Fermat's Last Theorem by BBC
   

1. ​
   
   Dangerous Knowledge by BBC
   

1. ​
   
   Hunting The Hidden Dimension by PBS
   

1. ​
   
   Movies/Series:
   
   A Beautiful Mind (2001)
   

1. ​
   
   
   Good Will Hunting (1997)
   

1. ​
   
   
   Proof (2005)
   

1. ​
   
   
   21 (2008)
   

1. ​
   
   
   The Imitation Game (2014)
   

1. ​
   
   Numb3rs
   

1. ​
   
   
2. Some topics to start with are Mathematical Logic, Real Analysis, Set Theory. Some of the good books are:
   
   Principles of Mathematical Analysis by Walter Rudin
   

1. ​
   
   Analysis - I by Terence Tao
   (The section on proof techniques and logic is sufficient for a start)
   

1. ​
   
   Native Set Theory by Paul Halmos
   

1. ​
   
   Introduction to Mathematical Logic by Elliott Mendelson
   

1. ​
   
   
   

I think you should not worry about the applicability right now. You will have to give it time. Once you get to know (really know) mathematics enough, you'll have no problems picking the higher level subjects. The foundations in mathematics are like tools; the more you understand how they work, and the more time you spend with them, the more you get proficient in the art of problem solving. It will come naturally, as you acquire the skills to formalize a problem and crack its solution open!