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Understanding that game development programmers are typically the most overworked and underpaid people in the industry, I would start off with a computer science degree from a "normal", ie not game specific, university. Depending on the type of games you want to work with, find games of that genre that have been open-sourced and work through the code. Then start writing small games and build your portfolio. Having a degree without anything to show for it will not get you far. Building your portfolio is the most important step in breaking into the system.
Excellent advice from lando there, fully agree.
Thx lando but wjo is the most payed in this industry
Ah, I see you mean who are the most highly paid programmers. The highest paid programmers are financial programmers on wall street, but also require heavy math skills and economic skills. For general programmers, an entry level programmer in normal corporate job should start around 50-70k at this time. The top game programmers can make in excess of 6 figures; however, these are people at the top of their game and only a handful make this kind of money. If you check dice.com, ie one of the best places to look for professional jobs in the industry, you'll see most programmer jobs are listed at 90-110k a year; however, the list of skills is far beyond what a normal programmer will have. They do this to get more applicants because of the good pay, but then they discount the pay because the programmer they hire does not have all the skills. A typical game programmer will start around 35-55k a year and then they will expect the programmer to put in over 60 hours a week and over 80 during the final months of the project. Whereas in corporate you will make the same but only work 40 hours a week. If you go with a start-up company, typically your wages are the same but you are expected to put in a lot more hours than you would in the corporate setting. Almost all game developers are start up companies or run with that mentality. Do a web search on game programmer working conditions, pay attention to the articles listed for EA games, and then realize that they are now considered one of the best companies to work for. If you do decide to go into games, a few years in the real programming market will give you better employment when you switch to games, more experience and will keep you from being taken advantage of, at least a little bit when you make the switch to games programming. It should be noted that going from the business community to a job in the gaming industry isn't too difficult, but if your degree is in game programming, trying to go to the business market is a lot harder.
Can I ask, what makes you keen to work in gaming? It does seem like a thankless task. For some reason, I always imagine it'd be uncomfortably hot. I guess they've got air conditioning, but I just can't shake the idea that it'll be dry and hot all the time. I dunno, all the best with it though if that's what you'd like to do.