This is good advice in today's landscape (an alternative approach to my answer above). Tech matters to me, I wanted to understand many things deeply, I didn't want my knowledge to exist entirely inside of someone else's genius work (such as Unreal Engine game development where you're scripting everything/wiring up game logic/etc.). However, doing the kind of thing I did a decade ago is almost impossible with the tech expectations of today, and also, simply not really necessary.
Also, know that the majority of game developers stay in the industry for only a few years (it's not always the dreamy job it sounds like). And an extremely high percentage of games (I have heard 90%+, I would say around 95% at least if you count all of the small titles) are not profitable. If you're still interested, I wish you the best (you're in for a lot of interesting work)!
I was tired of hearing "we want people who have shipped at least two titles" (classic chicken and egg problem) around the year 2004-2006. So I took 10 months off to create a game/base game tech. I targeted the Game Boy Advance. I created everything but the compiler and a small utility that smacked an "I'm GBA software" header on the ROM. This included the asset pipeline, audio mixer, even the interrupts library. What this allowed me to do was hand my project to people during an interview, and when they asked what I did, to say "everything you see there, as well as the things you don't such as x, y, etc.". Finished portfolio projects are invaluable in this space -- so many people have unfinished projects/demos that having something finished/polished really makes you the exception (it shows extreme dedication/diligence -- see the ninety-ninety rule). You might not need to take it to the extreme that I did, but it's nice to avoid situations where you have as portfolio work only projects worked on with others for which you don't have very clear details on what exactly you did (otherwise, people will have little confidence in what parts you are actually responsible/deserve praise for, if any). Don't show up to a gamedev interview with only a binder of dragon drawings expecting a job. And understand that a lot of the educational programs at "tighten up the graphics on level 3" schools are almost exclusively in it for the money (vs. a sound/working skillset for you, as so many people are interested in the idea of game development as a career). If you're going to do the university thing, get strong foundations in computer science (not necessarily deep math or super-advanced algorithms, but sound foundational knowledge and experience). Don't try to skip the basics (C matters).