Are APK files saved on my device when I install from the Play Store?
Try ES file explorer

Under settings select App Manager and select all apps. And select backup on the action bar below.

A backup of all apps apk is created to your SD card probably under backups folder(or check /set backup folder on your es file explorer, settings-> directory settings )

Now you've got all apps apk.. You can either transfer them through Bluetooth/Wi-Fi /USB connection

You can select system apps too if your phone's rooted