Step 1: Get the parts
See the Teagueduino bill of materials for part numbers and links to order from DigiKey. Get everything in the first section, and note the quantity of each (for example you'll need 10 connectors, and 5 LEDs).
Perhaps the trickiest part is the Teagueduino PCB. We're hoping to make blank Teagueduino PCBs available in the future, but until then feel free to make your own Teagueduino PCB from the Eagle files.
And if you curious what's happening electrically, take a look at the Teagueduino circuit schematic.