I'm looking to make a relatively simple game using solely graphics primitives (Arcs, Lines, Polygons, etc.).
I started doing this in C# by drawing to a Panel, but right now I'm hung up on how all the scaling works in terms of keeping the proportions the same when changing resolutions. Does anyone have any advice and / or tips on how to do something like this?
Not really a helpful answer but, don't use WinForms!!
If you want a good gaming platform, use DirectX, or XNA Game Studio.
There are two options:
1 - Scale everything so that it is sized at a certain percentage of the screen/window. For example, if you want your object to be 1/4 of the screen, then it's width is ScreenWidth/4 and height is ScreenHeight/4. The problem with this technique is that a screen's aspect ratio may make things short and fat or tall and wide. Usually this is addressed by determining one dimension and then using the screen's aspect ratio to determine the other dimension. Ie, Width=Height*AspectRatio.
2 - Make everything the same physical dimension. For example, you may want an object to appear exactly 1" by 1". You can use the screen's resolution (dots per inch) to scale your drawings accordingly. The problem with this is that while it may work well for 'average' sized screens, images may be too small on large screens or too large on small screens.
Most games use technique #1 (with compensation made for the aspect ratio). AR was not always a big deal, but now with widescreen monitors being so popular, it's almost required.
Also, like Richard said, WinForms is not great for games (except minesweeper!), but probably okay for teaching yourself.
You can do this using GDI+, and transforms. For details on using Matrix to do transforms, see this article on CodeProject.
That being said, this is much, much simpler using WPF's drawing options. In addition to being a retained mode model, which is much simpler when doing simple graphics (ie: move an object instead of constantly redrawing), it has some other nice benefits. The main benefit is that everything in WPF is done using floating point values, and is completely scalable, with no extra real effort. For details on this, see Shapes and Basic Drawing with WPF, which includes both drawing and transforming of shapes.