Planning

1. Decide on the placement of the microcontroller. That will determine the battery, the snaps that hook up to the dress and the starting point of the LEDs. On the waist line to the left is the recommended placing in order to be as close as possible to where the belt will be on the dress.
2. Decide on the pattern of the LED strands. They can spiral around the petticoat, swirl around semi randomly, zig zag up and down, whatever. It's recommended to connect them with a whole bunch of safety pins first before sewing them in place. In this project, the LEDs spiral around and around the petticoat. The dress was shorter than the petticoat so I the LEDs spiral finished up before they crossed over the edge of the dress.
3. Decide on the placement of the snaps. For the petticoat to dress snaps, use the larger magnetic buttons. You can go with either 4 snaps for just the color sensor or 6 snaps for both the color sensor and the on/off switch. You need to space them to make sure you can create separate connections that don't cross each other. And plan the order right so that you have direct lines to the place where the belt will be connected.

Sewing

It is extremely recommended to test all of the time! There's nothing more annoying that having a project not work and try to trace all of the connections to see where there might be a problem. 

Fun fact: Alligator clip to the magnetic buttons so they easily connect for testing.

Sew in the microcontroller with some regular black thread to keep it in place. Then start with the LEDs strand's connector. Using conductive thread connect the GND, D1 and Vout on the right to the wiring the connects to the strands.

Next sew the snaps to the GND on the right and the SCL (A1/D2), SDA (A2/D0) and 3Vo on the left. Use both the front side and inner side of the petticoat to avoid crossing circuits. You can later sew ribbon on top of each connections to further ensure they won't touch each other and also to avoid irritating the skin on the inner side. There are different techniques for sewing with conductive thread. You can read more about that here.

Sew up the LED strands. In this project, I sewed around every 5 LEDs either the LED itself or the wiring. Try to follow the natural curves of the petticoat so that the LEDs with flow nicely under the dress. What's nice about the Adafruit LEDs strands is that they barely weight anything and don't burden the petticoat.

For the battery, to have control over resetting the color, plan for a switch closer to the belt on the outside of the dress. To do this, cut the red wire to the battery and split it between 2 snaps. When the switch is "on" it essentially connects these 2 snaps and closes the circuit.

Lastly add a battery pocket that is held secure with a non conductive snap to keep it from moving around. There's a nice rubber band trick to keep the wires from detaching from the battery. Its in nearly all of the Adafruit tutorials, for example here. Make sure you choose the right battery size before sewing the pocket.