I would start with Arduino. It's the most common platform for this type of thing. There are plenty of online resources. Adafruit is a good combined resource and store. At minimum, you'll need:

The lights: maybe this is a good start https://www.adafruit.com/product/1461

An Arduino: https://www.adafruit.com/product/3345

A wifi or ethernet shield (a shield stacks on the Arduino board and add a function): https://www.adafruit.com/product/2971

A power supply: https://www.adafruit.com/product/658

You may also want a soldering iron and a nice case if this is a permanent project.

You could start with a raspberry pi, which will have a lower investment, but the Arduino's simplicity is great for learning.