How would I create this kind of circuit?
The circuit that you show in your sketch has three branches connected in parallel - something like this (although I only show two branches):







simulate this circuit &ndash; Schematic created using CircuitLab

In this circuit, SW1 is the main power switch - it turns everthing off.

SW2 will turn D1 off and on, if SW1 is on.

SW3 turns D2 and D3 off or on, if SW1 is on.

The resistors R1, R2, and R3 limit the current through the LEDs to about 13 mA.

As Respawned Fluff indicates, some more complex circuitry is required to drive the 7-segment displays.