The SNES pinout and board layout can be found here (the connector pin out is wrong on this link however):

http://www.gamesx.com/controldata/nessnes.htm

Correct connector pinout here: http://www.gamesx.com/controldata/snesdat.htm

The SNES bit order is here: http://pinouts.ru/Game/snescontroller_pinout.shtml

As far as the registers a great tutorial on how shift registers work can be found here:

http://www.protostack.com/blog/2010/05/introduction-to-74hc595-shift-register-controlling-16-leds/

The button leads on the 4021's needed to be pulled to ground, the mBed will not go LOW so I had to use pn2222 transistors to pull the pins to ground. (signifying a button press)

Here is a great explanation of small bipolar transistors such as these.

http://benkrasnow.blogspot.com/2011/05/how-to-design-transistor-circuit-that.html

I just used 10k ohm resistors(what was on hand) for the BASE to mBed and 10K ohm resistors from EMITTER to GND. (I know, sorry, and it works just fine.)

As it turns out, there are many different ways to do this. At first I was having a dumb moment thinking that the mbed could only turn pins to HIGH or 0. I wasn't declaring the pins as an output, which you must do if you want the mbed to be able to pull the pins to GND also. This is why I made such an elaborate transistor array. You do not have to do this! In fact, I ended up going straight from the mbed to the shift registers. This is the way you want to go. The third and final way of course is to have the mbed itself act as the shift registers. I have the start of the code for an example but I did not finish it so it's just in concept form.