Question: I am currently working on an application in WPF/C# for personal use. I am not a "classically trained" programmer. Simply a hobbyist that likes to code in his spare time. Is there any accepted approach to the progression of application development? I. E.; Make it work, add fault tolerance, create a gui, then performance optimization. Or maybe should I design the entire GUI first? Basically I am going to start a new project soon and would like to have some sort of "every program needs this" checklist.

Answer: 
If this is something you're doing as a hobby, you can choose any development methodology you want.

Typically, for commercial development, there's some form of prototype made first (for WPF, SketchFlow with Blend is fantastic for this). This is typically required since you most often need to "sell" the concept, either to a client, management, etc.

However, if you're doing this on your own, you have the freedom to do things any way you wish. One comment, however. I would, personally, not thing of "fault tolerance" as separate from "making it work", however, since "fault tolerance" comes with testing, and unit testing as you go is much more effective...
I try and take a look at my audience for the application. Then I'll create some use cases where I try and figure out what and why my application should even be built. Sometimes evaluating what you're writing and why you stumble across a lot of ideas that might shape the way you architect the application.

From there you can do as others suggest and put more importance on areas of focus.