|I have a 20 Mb modem, whatever that means.|
I have no idea what that means either. I suspect it means the network interface on it (ie: the RJ45 port you connect to the router with your network 'patch cable') is rated for 20 Mbps of bandwidth. However, it would be easy enough to find out if one googled the specs (specifications) of the actual device. A search along the lines of
should give you the info you need to verify what that means.
Definitely connect as follows:
Modem >> Router >> Computers
The router not only does NAT and DHCP but it also offers a firewall to help prevent intrusion. Even if you only have one computer accessing the internet, this is preferred. If you'd like more than one to access the internet, this very much the best way to go.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.