A router creates a subnetwork that cannot be accessed from outside (except through whatever exceptions you configure), and that presents to the outside as a single unit. Typical residential internet connections require both of these features -- you want your network closed off to the internet for security reasons (the internet is a scary place), and moreover your network must present itself as a single device because most ISPs only provide room for a single device per subscription. Which can be either a computer, or a router with any number of computers behind it (by far the most common arrangement for obvious reasons).
You are trying to create a second isolated subnetwork in your home network, which your existing router can do; but your ISP still needs your connection to only have a single apparent device on it. So, a second router.
A switch allows you to connect a larger number of devices to a subnetwork. It can enlarge a subnetwork, but it cannot create one on its own. You'll need one if you want to connect more (wired) devices to your PC#1 network than there are ports on your existing router. Or if you want to connect multiple devices in the same room that doesn't have the router in it, and you only want to lay one cable between the router and that room.