Welcome to the forum.
You can use an Arduino to control transistors to control power to the strips.
Connect the +of the LED strips to your positive supplies.
Connect the - to Logic Level N-Channel MOSFETs.
Drive the Gates from the Arduino, the MOSFETs will connect the - of the LED strips to the Ground of the supplies.
Connect all the grounds together.
Uno will do fine, lots of pins available for buttons.
