Justin,
You have already picked a path "post MS" and the path you picked is called "Amazon Services" which is just nothing but a name branded, very modified Android version.
If you want to add the full Play Store to your Amazon device, you can do that, but the instructions vary according to the actual device.
Here are the instructions for the stuff that was on sale this past week, yours is likely a generation earlier so you will need to ask the web more specifically for "correct to the version" instructions. They are out there for the viewing. YOU MAY NO'T WANT TO DO THIS BECAUSE IT CAN BRICK THE SYSTEM YOU ARE USED TO USING. Odds of you bricking your tablet are sorta high, as this isn't common level stuff for casual users.
http://www.thinkcomputers.org/how-to-install-google-chrome-on-the-amazon-kind...=======================================
NEXT point is
why are you asking?
You seem happy using what you are using and Amazon is never going to squeeze your balls for upgrade money like MS does. They will tempt you with a new tablet, however. For Cheap no less.
You are current living proof that someone can live post MS and be quite oblivious and quite content and happy while doing it .....Beware, you are also a fish that swims in strange water, you live inside a custom OS called "Silk" which includes a modified version of open source Chrome browser and a very modified open source Android version tying it all together. So yes, in a sense you use Chrome already, but a heavily modified Chrome.
Just use your tablet, you are already protected from all MS's weirdnesses.