I am rewiring a floor lamp. The lamp has two switches; one is on/off for the main center light. The smaller switch has three wires and feeds three smaller lights. The two sets of lights are independent of each other. The sequence for the three smaller lights are: Off, #1 on, next position #2&3 on, next position all three lights on and finally one more turn of the switch and all three lights go off. How do I wire this contraption?

Somewhat easy.

First, the ribbed side of the lamp cord is neutral and is connected to the silver screws. This will be all of the threaded sides of the lamps. All of the threaded sides of the lamp holders are connected together and go to the ribbed side of the cord. This ribbed side should go to the wider prong of using a polarized plug.

The center lamp is easy take power in and pass through the switch and to the center socket.

You missed telling me what the switch terminals are: I'll call them L, #1 and #2. Thus Pos #1: Off ; Pos #2: L to #1; Pos #3: L to #2 and Pos #4: L to #1 & #2.

L heads to power in from the wall.

#1 heads to the center of the lamp controlled by pos #2.

The other two lamps are wired in parallel. The center pins connected together and connected to #2 of the switch.

No magic.