... Why can't I just click on the color I want, type with that color until I'm done, and click on the next color I want to use?
Because, that isn't how Word works. When you click into the middle of existing text and start typing, Word takes on the style of the text you clicked into. That includes the formatting.
If you turn on tracked changes, you can get different colors for your changes. (See
Track Changes )
What you are doing is the primary purpose of this feature. You'll find it under the Review Tab.
You could create a custom character style if you wanted and put that on your QAT and/or assign it to a keyboard shortcut so that when you start typing you could simply click on that icon or use the shortcut to apply your custom style.