There's Getting Started with Arduino and we've also put together a list of AVR resources. There are many, many other resources out there as well. Depending on where you are located, there may even be classes at your local hackerspace.
When you are programming any arduino compatible device such as Peggy you connect your computer to a programmer (such as a USB-TTL cable) and your programmer to the device. After it is programmed you can disconnect everything so that the Peggy runs its program on its own. The exception to that is when you are using the computer to send data to the board (as in the case of the video Peggy).
As far as the interactive LED panels are concerned, you don't need to know anything about programming, as they are completely analog and cannot be programmed.
Yes, you could use the interactive LED panels for a floor system, but you would need to work out a support structure for the surface you would use over the panels. We normally recommend clear glass or plexiglass which can be sanded/frosted if you want a diffused surface.