You made about 5 steps at once. Do this step by step. You need for example a few test sketches, before trying to run the final project.
Is the Arduino board running, can you upload a sketch to it ?
This is your board:
Can you make the led of the Arduino board blink when it is attached to your board ?
Check the wiring of the shift registers. Either the schematic is wrong, or you have connected them wrong.
This is explanation and shows cascading for 2 shift registers:
http://bildr.org/2011/02/74hc595/
You have enough pins, so you don't need cascading.
If the shift registers are working, you have to check how the ULN2004 are wired. The connectors to the leds have no power pin but a GND pin. The ULN2004 is like an open-collector output, you need a power pin for the leds.
What do the leds look like ? Do they have a series resistor build in. I don't think so. You can't connect the leds like that.