I have a 4 slot battery holder with 2 wires coming out of it, best I can describe it. I'm planning to get a 14.8V Li-ion PCB charger (of course this will be used with 3.7V Li-ion batteries).
One thing I don't understand to do is the connection, as in do I just connect one wire to the positive end and the other to the negative end? Or should I instead get a separate 4 slot battery holder and make the connections like this? https://gyazo.com/fa6e1f7c2dd25f4c580b62b9053dde22
How you connect the battery depends on the charger used.
The example image that you give probably shows connection to a battery management system which as well as charging provides protection against over voltage, undervoltage and over current discharge operation. This is a very common method of connection and is recommended.
I acquired a number of multiple cell Liion battery packs produced by a major vacuum cleaner manufacturer (D----) which had no means of balancing cells. They suffered an early death - as should have been wholly expected by the "designer",
Note that LiIon cells usually are rated at 4.2V fully charged with 3.6 or 3.7 V representing the mean voltage during discharge. The 4 cell battery MUST be charged with a properly designed charger if it is not to die an early and possibly flaming death.
Without that PCB board with wires from each 3.7V battery you may accidentally discharge the batteries so deeply that it will damage them. Maybe your battery holder has a PCB protection board? if no, then you need one.
You need:
Some such PCB also provide balancing circuitry, which makes sure that all 4 batteries have the same voltage. Without a balancing circuit overtime one battery can be fully charged at 4.2V while other batteries may stay undercharged at 3.9V for example. (of course, if you charge every battery separately this will not happen)
