Once Puppy is up and running,
Click on Menu -> Setup > Puppy universal installer.
Make sure your USB Memory stick is plugged in.
Select USB Flash drive option and follow the prompts
It's a good idea to use the http://www.pendrivelinux.com/2007/09/17/testing-your-system-for-usb-boot-compatibility/ utility to make your USB drive bootable before attempting to install Puppy. I had a USB memory stick that didn't want to boot with Puppy until I had run a boot utility like this on it (HP also have a USB Flash drive boot utility).
Once the universal installer is finished you can reboot your machine. Don't save a Puppy Save file. Boot with the Flash drive, choose your keyboard, graphics etc, again and continue...