At a super broad level, there are three things a good designer should have:
- Very good communication skills
- Basic scripting/concepting ability
- Detailed knowledge of game systems and a proven track record creating great experiences
A computer science degree would definitely help with the scripting, and one of the best ways to get into design is to, well, design things. Pick up a free engine like Unity or Unreal Engine 4 and start tinkering, try out mechanics and such while learning how the engine functions. Eventually, you can start making really simple games and getting critique from people, and the more you do this the better your games will get.
Another thing: Internships. If you are able to build a basis where you can show employers a portfolio of what you've created, that's a great way to get into internships. This would give you some applicable real-world experience, and unlike a lot of professions Game Design is almost exclusively about experience rather than credentials.
TLDR: A good path for you might be to get a Computer Science degree while pursuing video game creation in your spare time. Depending on the college there might even be targeted courses you can take, to help jump start your ability to script and create games from scratch.
