Lets break this up into manageable parts
I have reordered them a little bit
I think I would need to hook the cable modem to the cable jack, run an Ethernet line from the modem to the input of the splitter. I think I would need to then run an Ethernet line from one output port to my pc, and the other to the wifi router. is this correct?
That would most likely lead to a non functioning network.
Ethernet is intended for connecting two single points together. It doesnt have any mechanisms to handle more units talking on the same line.
I would like to be able to use one or the other at one time...not both. can this be done?
Is there any reason you don't want to use both at the same time?
From a practical point of view most routers have a switch built in so you can connect more than one device by cable and at the same time have users connected on the Wifi.
This is how I would recommend you do it.
when in the wifi selection the laptop should be able to find the signal right?
Yes. It will find Wifi signal as long as the router is powered up and/or the Wifi function isn't switched off from the routers menu
is there anything I need to consider when getting a wifi router? I would only need a signal to reach from one floor of the house to the other.
If you buy cheap stuff you will get cheap stuff. It might work, it might not.
I would suggest you buy something with the AC Wifi standard in it.
This would be a good choice if you can afford the price:
Or any of the Asus RT-AC series
This site has fairly unbiased reviews afaik.
also how do you secure the router ie change admin password?
This is usually a part of the setup routine of the router. You choose a new admin password, set the SSID and the Wifi password, perhaps you adjust the clock and some other settings too
If this isnt part of the setup routine you will have to log in to the router and manually set these things.
How to do this will differ from router to router.