Are APK files saved on my device when I install from the Play Store? I used a Nexus 7 for a couple of years and got a Nexus 10 earlier this year.  I installed a number of kids books/games on the 7 and wanted to transfer them to the 10.  Now I can't find some of them at the store.  Would the APK files still be located on the 7?
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