A game designer is loosely analogous to a director in film and TV. You would be responsible for envisioning the project and seeing it produced. The design of games includes two broad aspects - the ludological design, i.e. the design of the gameplay mechanics, and the narratological design, or the design of the narrative elements.

Game design is multidisciplinary, and in smaller studios, you may see an overlap in roles where the designers may get involved in the art or programming elements (as per their skillset) as much as you'd expect of an artist or a programmer. Working with teams of specialists is an important skill, as is communication. The standard way in the industry is a game design document (you can find those for some classics like Doom or the first GTA online). This is the game industry equivalent of a software design document and mentions anything your team may need to you. You could draw a parallel with a TV series bible, except you'd almost always find some notes on implementation and much more computer science stuff than a TV series bible, especially where you consider target platforms.

• This list is broad, but I'd list the following skills. You could lean more towards the art or coding side of things, but I wouldn't recommend leaving the other out entirely. Have a basic familiarity with each, but have something that you can excel at:
• The ability to design gameplay mechanics and the storyline for the game. You don't need a full-blown technical understanding of it, but knowing psychology on a practical level helps a lot. Games that appeal to our minds on a psychological level have a different pull. Consider the urge to complete a half-finished puzzle, or the urge to compete for the top position in a leaderboard. Some games appeal to our desire to explore and play around with things. Others reward attempts to break the game by thinking out of the box. If writing is your thing, focus on the ability to deliver emotional experiences. In my view, high-concept ideas like saving the universe or averting a thermonuclear war are all noble pursuits, but it's stories about the human experience that we relate to, remember, and even come back to.
• Technical proficiency. Know the basic programming constructs (conditionals, iterators, classes and objects). For performance reasons, C++ is the undisputed language of choice for game programming, and is supported by many game engines, including Unreal (4.x onwards) and CryEngine. C# is used to script in the Unity Engine, which has a significant presence in the indie scene. If coding is not your thing, I'd recommend knowing enough to put together a rough prototype in an engine using visual scripting. Both Unity and Unreal support it, and the visual scripting language of Unreal is just a visual interface that lets you work with the engine's C++ code under the hood. Also, you should know about version control. All software development uses it.
• The ability to work with 2D and 3D art (as the project may require). If art is your thing, you should aim to get familiar with software like Photoshop which are used to process game assets. For 3D art, 3D modelling, texturing, and rigging would be a major part of what game artists do. Also, always know how to work with imported assets in game engines. If mathematics isn't your thing, there's also bad news - you need to know some for this. The good news is that it's mostly simple. Knowing your way around in 3D coordinate spaces and the concept of surface normals can save you from more than a few 'gotchas'.
• Teamwork. Game development is highly collaborative. You need verbal communication and written communication skills to discuss ideas with your team and record them. Communication also involves the ability to pitch your ideas for funding. Look up the concept of an elevator pitch for good rules of thumb. The standard procedure is scrums - iterative development and testing. Teams meet regularly to discuss goals and plan out their work.

My last tip would allude to real world concerns. Make sure you can work with deadlines and look adversity in the eye and rise up to the challenge. Although it's true of art as well, just like all writing is rewriting, almost all of coding is debugging. You may have to spend hours trying to squash that one bug with no idea how it got in only to find a small typo somewhere in your code after running 793 different tests (I'm just making that number up, but you get the point). Staying motivated, focused, and not losing your sanity can be a greater challenge than debugging itself.

None of which is meant to discourage you. Work on gradually developing your skills. Put together a few small-scale projects to learn by doing. Unity and Unreal have some excellent playlists to guide you through. The only tip I'd give you is to take every opportunity you have to make the project your own, innovate in some way, or modify some element that doesn't quite feel right. As a game designer, you should try to introduce changes that alter the gameplay in a significant manner. Don't limit yourself to easy or difficult. Focus on making it different. It doesn't always have to be a major overhaul. Portal transformed the FPS genre into a puzzle by one simple, but fundamental change - you ‘shoot’ portals to teleport rather than bullets.