Question : VLANs to connect two networks?

I have two outside networks (WANS) that I'm trying to access through our office network.  One destination is a cable modem to the Internet, and the other is a client's office through a DSL modem.  Users in our office will need to access either, but the client can't be able to see our network or use our Internet connection.

I have an understanding of networking fundamentals, but have never done anything hands-on with Cisco routers before.  After lots of reading/research, I decided on the 2811 and added a 4 port switch.  I thought each network would plug into the one of the switch ports and our network would plug into one of the "built-in" ports.  Here's the basic setup envisioned:

                                     Cable Modem(Internet)          DSL Router (Client)
                                                          |                        |
                                              172.16.10.1 - |        | - 192.168.1.0
                                                                  |        |
                                                                Cisco 2811
                                                                        |
                                                             Internal Network    
                                                                   10.1.0.0  

I know I've got a steep learning curve with the routing tables and access rules, but I'm slowly making progress.  The problem I'm running into is how to have the two networks identified by the switch ports.  With the built-in ethernet ports, I can assign an IP address and have that successfully working using DHCP.  However, with the switch ports, it appears that I make these as access or trunks, and then assign the ports to a VLAN?

This goes against my understanding as I thought VLANs were for dividing a physical network into smaller subnets, not that each network would need a VLAN assignment.  

I'll go with the assumption that I'm misunderstanding something.  How do I get the router to recognize both networks?

Answer : VLANs to connect two networks?

There you go.  

Although, I would use the two layer 3 ports for the Internet and Client VLAN and use the switch side for Internal.  This will allow for easier routing of traffic from the Internal network out to either the Internet or Client networks.