I think the only confused person here is me

Yep, a four port broadband router shoudl work for you.
A four porter will have five ports on the back. Four for your network devices and one to go back to the 192 (or internet provider) network.

Using your diagram, the Linksys will have two addresses, one on 192 the other on 10.

The (say) Linksys allows you to configure its own IP address and will pull all the DHCP info it needs to connect to the 192 network.
All you have to do then is set the default gateway on your pcs to match the address you have configured on your router.
The Linksys will then translate (NAT) your 10. network over to the 192 network.

Honestly, it sounds more complicated than it is to do!

One last thing, don't forget to setup the DNS entries on your 10. network!
I took the easy way on my setup. I setup my local network DNS server to refer to the ISP DNS server, then I didn't need to change my PC setup.