I have no experience developing games or programming and a basic understanding of computers at best, but I know I want to explore the field of game development (or design or any field that involves hands-on work with games) and that such skills are essential to doing so. How exactly should I begin developing them? Is there a particular skill that's better to start with than others? When I consider the myriad of things I'd need to know, it often feels overwhelming, but I have no idea how to break the task of learning them down into more manageable pieces. Any help with this would be immensely appreciated.

Start learning a programming language (C++, java,...). Once you know one of those quite well you can start creating 2D games with a game engine that uses your programming language. Say you're doing C++ you could make 2D games with SFML for example.
After you've mastered creating 2D games (try out multiple frameworks by the way) you can try making 3D games with, for example, Unity. Unity is to my knowledge the most accesible game dev software.

Generally speaking, there is the art side and programming side. As for platforms there are the consoles, the PC, the Mac and mobile devices like the ipad, phones etc..

For the consoles and PC you generally use the C/C++ languages (personally I found the Sams teach yourself C++ in 21 days pretty good to start). And for the mobile devices you have Andoid OS(Java, C++, Opengl), Windows phone OS(C#,VB, C++, Direc3d) and IOS(Objective C = don't know much about IOS,maybe Opengl) and the rest of the non-OS phones are generally in Java.

So it really depends on how serious you are and how your ability to learn and apply a language. C/C++ can get pretty tough when you get into the win32 API, Opengl and DirectX, plus you have the maths side of things. C# is a popular langauge just now and significantly easier than C++.

I'd recommend a nice introductory course to programming like the JavaScript one on Codecademy then you'd have enough coding knowledge to dive head first into Unity tutorials (I recommend the Walker Brothers ones).

I agree with greyfox. I hadn't looked at GameMaker for quite some time, but just visited their homepage, and you can actually make some really cool games with it, like this one:

https://www.yoyogames.com/showcase/38

If I, with all respect, had as little experience with game development as you apparently do, I would totally start with this program! And who knows, if you become good at it you may not even need to learn anything else (in which case you should consider yourself lucky taken into consideration all the headaches you will be spared from learning a more "serious" language like C++)

Thank you! This will all be extremely helpful. I appreciate it!