How can I add lights and a switch downstream of a pull-chain fixture?
So your existing fixture (assuming standard US wiring, you'll have to verify) has a hot (black), neutral (white), and ground (bare) and has constant power.  The additional lights will need 4 wires - constant hot to travel to the switch (black), a neutral (white) and a switched power that will come back from the switch (red).

The existing wire (assuming 14G wire for lighting circuits) is sold as "14/2" cable.  The ground isn't counted in the number scheme, so it's considered to have 2 conductors.  What you will need for the rest of the run is "14/3".  This new cable (black, white, red) will run from the existing light to each of the new lights and then to the switch box.

At the existing light, black connects to black and white to white, but the red will just be capped off and unused.  Red will be power coming from the switch.  In the future if you wanted the pull chain light to be controlled by the switch, you could hook that light's black wire to the new red.

At the new lights, white will connect to white and the black on the fixture will connect to the "switched" red wire.  The black wire of the 14/3 will not connect to the new light and will just "pass through" the box.  In each box, all ground wires are connected to each other and the bare or green wire from the fixture.

At the switch, the white wire will be capped (unless you use a smart switch that needs the neutral) and the black and red wires will connect to the two screw terminals on the light switch.

Older wiring practices would allow you to do all of this without the extra red wire, but now all switch boxes require a neutral so you must use 14/3 to do it properly and allow for future flexibility.  You could do this now with a mix of 14/2 and 14/3, but if you just use 14/3 you don't have to buy 2 rolls of different cable and again, it gives you flexibility later.

Diagram added by request:


The "dotted" red wire is optional and if you never want the pull chain light to be on a switch it will serve no purpose.  The white wire going to the switch is not used unless you have a smart switch but it is required by code.