Ideally, printers are to be seperated from their clients (and the internet) as much as possible.
There seems to be some kind of confusion on your side between categories of network devices. I'm assuming you're coming from a COTS standpoint which usually means that a device called "Router" does carry the jobs of "Router", "Firewall", "Switch", "WiFi Access Point", "Gateway" and "NAT", which might lead to the confusion.
A network architecture that can provide this might in it's simplest possible layout (which can easily be more complicated when deploying microsegmentation between clients and/or printers for example) look like this:
[Clients -> (Firewall] -> {Print Server) -> Printers}
where
[] = the network connection between the clients and your Firewall,
() = the network connection between the print server and your Firewall,
{} = the (network?) connection between the Print Server and your Printer(s)
Now, the connection between the print server and the Printer(s) doesn't need to be networked. The printers could just be locally connected to the print server, for example by USB or LPT.
Also worth noting is that this setup is minimal for this case; usually, there are different subnets of clients that need different printer access which leads to vastly more complicated setups that do seperate networks even further, using more Firewalls, Gateways and Print Servers.
Please also note that none of this includes internet access thus far. In this setup, internet might be provided through the Firewall (over a seperate network connection) and access rules on the firewall would prohibit internet access for the print server.
Additionally, COTS "Router" devices usually do not provide all features of actual Firewalls, making this setup hard to achieve with it - and you usually do not. One notable exception are COTS Routers that offer a USB Port to connect a printer to in which case the device itself functions as the print server.
Lastly, please note that the threat model for private households and companies vastly differ and this setup might not be suitable for home deployment.