The first thing is will the networks need to be able to cross-communicate in some way? In other words, will you have to send network traffic from one network to the other, such as monitoring software, printing, or domain control?
You would need to get an additional router. Your Qwest modem is going to be the primary internet connection, and it should have two routers connected to it only. One router will act as the default gateway for your private network, the other will be the default gateway for your public network.
Even though the above configuration does give you two separate networks, both with access to the internet, it doesn't really block network traffic between the separate networks. You need a firewall for this, and with a firewall you could use just a single device instead of three routers.
Given what you have described, it is hard for us to tell what to recommend, what you need, and what issues you might come up with. A little more detail on your situation and what equipment you might be using would be helpful in getting a more detailed response.