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So to start off, it's great that you know where you would like to work. Treat that as your end goal. Study up on all the games that company makes, and try to start designing and creating for similar types of games. If you create cute cuddly casual games, and try to get a job at sony who makes hardcore console games, you'll find your portfolio doesn't line up with that they are looking for and will get passed over. Create content that you know they will love to see in a portfolio.
Another thing is to consider what it is you want to do. You state that you want to design and animate a story/game. At a place like sony, designers design and animators animate.... so pick what it is you really want to do and dive head on into that. Don't let me deter you from trying to make a small game with a hand full of friends tho. I'd actually highly suggest that you do try design, concept, modeling, rigging, texturing, lighting, animation, Creating user interfaces and even scripting/programming to hook it all up. Get a full understanding of what it is to make a game and find what part interests you the most. That is where you find out where you will shine the most.
If you and a few friends can manage to get together and actually create a small engaging and impressive game that you can show off on a device, this will help you tons to get into a game company. The will see that you actually understand what goes into a game, and if it's successfully done, they will appreciate your skills so much more because there will be less to have to teach you.
As I said before, treat Sony as the end goal. That does not mean don't apply to them as soon as you are out of school. Heck, try to get an internship there. But don't be shy to reach out to every other game company for jobs and internships. Every game company gets you one step closer to your end goal of being at sony. Even if they turn you down, don't let that deter you. Create more and try again. Hope this helps. Best of luck to you.
100% of 1 Students