So first let's explore game Design versus code vs. Graphics , I think the best way to explain this is an architect who plans out a structure versus the construction team who builds it versus the team who paints it .
The Game Designers I have worked with tend to be people who Have a penchant for not just thinking about games but developing the full idea of a game . For example a Game Designer I work with was tasked with Creating a small title called GemJump . His job was to decide what the different platform types players would jump on in the game are, the unlockables that players could earn , figure out what sort of social features/ leaderboards we would have in place are, what players special abilities are, how long our target gameplay session was . And all of these things he could do with a pencil and pad of paper but he is a particularly good Designer so he'd write Game design documents, play other similar games to what we anted to make , and essentially craft the foundation of what the game was that we were making . While his main focus was the design he also enjoyed tweaking and crafting the particulars of the experience to make sure it played and " felt " right .
I am a programmer my job isn't that of a designer I do the building , if my game designer wants a character to jump a certain Height I program that character to jump or I create a tool or means for him to set the character jump height himself . I do the vast majority of this by writing code or essentially writing instructions for the game to carry out " when player taps screen , Jump!" . The code for that can be a little more complicated but that's the idea . The designer creates the idea of the experience and Programmers bring the experience to life putting in the features before handing the project back to a Designer to tighten and tweak .
Graphic designers( and I assume you mean 2D / 3D Modelers and artists) are another discipline they create the visuals of the game, see that Shiny enchanted shield with the flames running around it ? That's a skilled artists doing . See those trees blowing in the wind , an artist modeled that tree and depending on their knowledge may also have animated the swaying movements . The artists create the visual elements of the game They are the ones who make the col cars we drive, the awesome Boss monsters we fight , or who make the tank Firing at you from across the battlefield seem so realistic.
These are just three disciplines that contribute to a finished game and there are many many more .
So now my question to you would be do you want to design games?
the easiest way to do that is to begin writing a journal about every game you play and think about what you love/ hate / and wish to improve about every game experience you have . if you have a Game idea I would encourage you to look up a Game Design document and write all the lovely details you like and once you finish pick up a simple game engine such as Construct 2 , Game Maker studio , Game Salad or another engine to help introduce you into making games.
Do you want to engineer Games? I would suggest you pick up a game engine to start with Game salad, Game Maker studio , construct 2 , and look up youtube tutorials on coding . Do something small first like a basic game where you click a moving object and get points for it . My very first game wasn't even visual ! it was a text based RPG and creating that helped lay the foundation for how I understood to code logical structures. as you advance some basic introduction to programming videos may help but I'm more of a hands on learner and I think the best way to learn is to build a game of your own ASAP .
Do you want to create Awesome art ? If you'd like to create visuals I would suggest studying some basic art theory , maybe a simple youtube tutorial on drawing and modeling and looking at Maya , 3D Studio Max, Blender, or checking out some of the assets being sold on TurboSquid. if you like 2D art you can look into sprite editing software ( I can't think of any great ones at the moment but there are plenty out there and I think Game Maker Studio and Construct 2 come with their own Sprite editors).
I know I gave you a lot of information but feel free to reach out if you need any clarification on what to look for and if I don't have an answer I can always reach out to some of my friends who have been making games for a while too . Good luck !