If you haven't received the update that allows Google Drive backups yet,
and you need to move your WhatsApp history over to a new device, it's a
pretty painless procedure. All you need is a microSD card and a few
minutes to spare.
- Insert the microSD card into the device that has WhatsApp on it
- In WhatsApp, go to Settings > Chats and calls > Back up chats
- Remove the SD card and place it into your other device
- Install WhatsApp on the other device
- WhatsApp should detect the backup and prompt you to restore it, which you should do
If you need to move the backup, use a file explorer to find '/sdcard/WhatsApp' and move it to the correct folder.
