Router troubles

I am trying to get a Linksys router to work with my two Macintosh
computers (350 MHz iMac G3, and 1.42 GHz iBook G4). My intent is to
have the iBook connected wirelessly via the router, and the iMac
connecting via Ethernet to the router. A cable modem is also plugged
into the router. So far, the iBook works just fine, but not once have I
gotten the iMac to connect.

When I have the TCP/IP set to be automatically configured in Network
preferences (on the iMac), an IP number does come up; however, it says
that I may not be able to connect to the Internet because it's a self
assigned IP number. (Also, it has NOTHING at all under Router when it
is set to get the numbers automatically.) I tried configuring the IP
number automatically and manually putting in the Router number, but I
still cannot connect with the iMac.

By the way, there is another problem I have: sometimes the wireless
connection on the iBook dies, and another network pops up in the
AirPort menu. (The network I hook to usually is called Linksys in that
menu.) My theory is that another nearby network is interfering.