Can I connect the Ethernet cable from my cable modem into
- A wireless router
- A direct Ethernet connection to a PC
Do I need anything special? What kind of splitter?
"Splitting" with Ethernet typically means "use a switch."
Anything you plug into any of the ports can talk to anything else that's plugged in.
However, your cable modem is a bit different than a PC or smartphone - if it's like most of them, it can only talk to 1 other device (NOT multiple devices) on its own, and that device needs to be a router.*
With your situation, since you need a router and a switch together. Order to connect is Cable Moden -> Router -> Switch. It's fortunate that your typical "wireless router" has a "built in switch" that's internally connected the right way.
Cable modem goes in the WAN port. Computers go in the LAN ports, or connect wirelessly. Wireless devices typically act like they are connected to one of the LAN ports, logically.
You can get standalone routers that require you to put a switch on the "LAN side" of them but since most people want wireless, it's not what people usually get.
*It can also be a single PC. It cannot be multiple PCs without the aid of a router.
Connect the cable modem to the wan port on a router, and then connect your other two devices to the LAN ports on the router.