Currently, my house only has one wall plate, which uses a cat5 cable that utilizes a wall plate with pre-installed wires (blue, green, white, black, red, yellow) that the cat5 wires also connect to. From it, I use a splitter to connect to my phone and modem. I am in the process of installing data ports at other locations within my house. I pulled the incoming cat5 wire from my upstairs kitchen to my basement where I have a switch. I am unclear on how to connect the incoming line to the switch and then to my modem and phone. Especially since both the phone and modem have female RJ11 ports.
Are you under the impression that you have an ethernet cable that just happens to supply phone service, but will also magically supply ethernet if you plug it into a switch?
You (appear) to have a 4 pair cable (if it's cat 5) which has probably (you're a bit unclear, but using the most likely scenario) got POTS (voice telephone, or plain old telephone service) and DSL using one pair - typically the blue pair - that's presently connected to a 3-line wall plate, but (probably) only using one line (typically the red/green pair in the center) since you mention using a splitter from that to your modem (presumably a DSL modem) and your Phone (presumably POTS.) The other 3 pairs are probably dead and doing nothing.
So, what you need wherever you relocate this cable to is to either reconnect it exactly as it was to the 3-line plate you had it terminated to before, and then to your phone (probably with a DSL filter in line there) and modem (without a DSL filter so it gets DSL), and then plug the other side of the modem into your router and your router into your switch, or if your modem is a router, plug the ethernet side of your modem/router into your switch.
The telephone side of this (POTS/DSL) has no connection to your ethernet switch.
If I'm guessing correctly, you only really need to connect the blue pair to an RJ-11 jack (Red/Green - center pair) and your splitter to your telephone and modem, or connect it to TWO RJ-11 jacks and have no splitter required. You can do that with POTS/DSL. You can't do that with Ethernet. Blue/White connects to Red, White/Blue to Green.
There are a variety of less-likely-these-days scenarios involving actually using two or three telephone lines, but they are very unlikely in the current era, and I'm having to assume a great deal, so I assume the unlikely things are not what's happening here.
Alternatively, put it back where it was, and run a cable from an Ethernet port on your Router or Modem/Router to the switch.