Since you didn't provide a picture, or a very helpful description of what you're looking at. I'll try answering your question by explaining how the switch itself works, which will hopefully help you understand the problem better.
Single Pole Single Throw (SPST) Pull Chain Switch
The pull chain switch that controls the light(s), is a single pole single throw (SPST) switch. It has two positions ON (Closed), and OFF (Open). Drawn simply, it would look something like this.
Switch shown in ON (Closed) position.
When the switch is in the ON (Closed) position, current is allowed to flow through the switch, through the light(s), and back to the the source (via neutral).
When the chain is pulled and released, the internal contact rotates 90° (1/4 turn) into the OFF (Open) position.
When the switch is in this position, current is not allowed to flow through the switch, and the light is not lit.
This is why the pull chain switch that controls the light(s) only has two leads.
Single Pole Multiple Throw (SPnT) Pull Chain Switch
The pull chain switch that controls the fan, is a single pole multiple throw switch. It has multiple positions, which allows it to control the speed of the fan. Draw simply, it would look something like this.
Switch shown in OFF (Open) position.
When the chain is pulled and released on this switch, the internal contact rotates 45° (1/8 turn) to the next position.
Another pull, another turn.
Pull again, turn some more.
One final pull brings the switch 180° around, and again to the OFF (Open) position.
By manipulating the output of this switch, the fan is able to whirl around at various speeds depending on the switches position. The number of output leads, will depend on the switch. How those leads are connected to the fan motor, will depend on the fan manufacturer. This simply illustrates the basic principle of how the switch works.
As always electrical work can be dangerous, never be afraid to contact a qualified Electrician
There is no way to tell what to connect to. It depends on how they wired it. I think it is safe to assume you should connect to the single black since it is in that box by itself like that. And yes, you would connect to the white and green with the other two wires. The instructions are not sparse, they show what you need to connect to. They do not know or care how your house is wired, nor should they. I will say, that box for the porch light is NOT appropriate. That is a collar box meant to mount onto an existing flush box. The wood of the logs is exposed to the back side of the box.
I also see a hole in the back of that collar box. Is this new light meant to be inside or out? If outside you'll need to replace that existing Wiremold collar box with a regular box, then run a cable from the back of the box into the back of a flush box outside.
Best Answer
It looks like you have a switch leg possibly improperly wired and possibly 2 lights or a light and power going someplace else like an outlet (possibly a spare) You will need to figure out which one is the switch then it will be easy, your main is labeled. When you find the switch cable it is probably the one with black tape on the white wire swap that black and white so the white is always hot and the black is now the switched hot. Then the light fixture can be hooked up to the black switches hot and the white from the main. If everything works the other cable may be a spare, if you find another outlet or light that doesn’t work the one that is not connected needs to be connected to the Main cable white and black. It would be good to put a wirenut or crimp sleeve on the grounds and attach them to the box as it is a metal box the ground should be connected to the box , this may also be done with the new fixture green wire.