Just download them for now, don't open them.
Here are the apps you need with links to download them.
All of which is to say, this process is relatively safe. Once these apps are installed, they will be updated and managed by the Google Play Store, downloading directly from Google's services. APKMirror is owned by Illogical Robot LLC, which also owns the Android-focused news site Android Police. You will be downloading all of these files from. You can find out which tablet you have at: Settings > Device Options > About Fire Tablet. Different Fire tablets run different versions of Android, so the files you need to install vary according to which Fire tablet you have. Here's where it gets a little complicated.
The first three are frameworks that the last one-the Play Store app-needs to function, so they need to be installed first. We need to download four applications to get the Google Play Store working: Google Account Manager, Google Services Framework, Google Play Service, and Google Play Store.