"Client for Microsoft Networks" is irrelevant here.  TCP/IP binds to
network adapters (like the WPC54G), not to network clients.

What exactly appears in the Winipcfg drop-down list -- I'd expect "PPP
Adapter" (dial-up networking) and at least one actual network adapter.
What network adapters do you see if you do "ipconfig /all" in a
command prompt window (Start | Run | command)?

Un-installing and re-installing the WPC54G and its software might
help.

I don't know if this applies to your situation, but I've had so much
trouble getting Linksys network adapters to work, over many years and
in several versions of Windows, that I won't buy Linksys for myself or
for my consulting clients.  If a client's computer has a problem with
a Linksys adapter and an un-install/re-install doesn't fix it, I
replace the Linksys with a different brand -- it's less expensive for
the client (and better for my sanity) than trying to figure out what's
wrong with the Linksys.  For wireless G adapters, Motorola is the best
that I've found, followed by D-Link, Netgear, and Belkin.

There is no entry labeled "TCP/IP -> Clienet for Microsoft Networks". In my
network configuration table, the topmost entry is "Client for Microsoft
Networks". The properties for that entry are simple; just a "General" tab,
and some settings. On mine I have selected, "Quick logon" in the "Network
logon options". I have no Windows NT domain servers, so I leave that one
unchecked.

Below that should be an entry with a green PCB icon labeled, "Wireless-G
Notebook Adapter v.2.0". Bringing up the properties for that one should
show a menu box with tabs; mine has three tabs, but that might be driver
dependent. I did not have to change anything on those tabs.

The "TCP/IP -> Wireless-G Notebook Adapter v.2.0" sounds like the proper
one for the adapter. It indicates that the "Bindings" tab on the adapter
properties is checked. Here is where things can go wrong. I have five tabs,
but I have seen fewer on other computers. My tabs are:

"IP Address"
-------------
Mine is set for Obtain an IP address automatically. If your router is
making DHCP assignments of IP addresses, this setting should work. Other
than that, I have the "Detect connection to network media" box checked. I
have flipped between checked and unchecked over time without seeing an
effect. I am not enough of a network guru to fully understand the purpose
of this one, so if an MVP says I am wrong, well, I must be wrong. :P

"WINS Configuration"
--------------------
Mine is set to "Disable WINS Resolution". I don't have WINS server. You
probably don't, either. If you do, you should have mentioned it. If you
don't know, it is unlikely that you have one.

"Gateway"
---------
There are no entries here. This should not be used unless you are setting
up a static LAN. If you are, just say so; setting this up for static IP
addresses isn't too hard.

"DNS Configuration"
-------------------
The same thing applies to this tab as for the "Gateway" tab. Unless you are
setting up a static LAN, nothing here.

"NetBIOS"
---------
This one should have a check in the box, and be grayed out. You are behind
a router, so disabling this is not necessary; and it *will* break the
TCP/IP LAN.

"Advanced"
----------
As far as I know, mine is at the default settings. The only property is
"Allow Binding to ATM", and it is set to, "No". Set this protocol to be the
default protocol" is checked.

"Bindings"
----------
"Client for Microsoft Networks" is checked; necessary if you are going to
access LAN shares from this computer.

"File and printer sharing for Microsoft Networks" is checked; necessary if
you are going to make any files, and/or an attached printer available to
other computers on the LAN.

"Microsoft Family Logon"
------------------------
Most people don't have this on a Windows ME computer, but I was playing
with setting up different users. It is probably a drain on my resources,
though, and does not afford any security. It mostly enables a single
computer to be shared by people who want to have there own Desktop, and
other such personalization.