Are you trying to connect to your PC through ActiveSync and the USB cable? Or through your ethernet card?
As long as you can get an ActiveSync connection between your PC and your Axim, and your PC is connected to the Internet, you should be able to access the Internet from the Axim. ActiveSync has a "pass through" feature to allow this. You may have to try switching the "Work" and "Home" connection settings between... "Work" and "Home."

You shouldn't have to "connect" at all; you should be able to fire up Pocket IE and start browsing. If you have a firewall, you may need to adjust the settings to do this.
If you're using your ethernet card, you'll have to configure it to connect to your network, which it should do automatically when you plug in the ethernet cable. However, you won't be able to access the Internet this way unless you set up a proxy or NAT server on your PC, or use a hardware router. A desktop PC by itself doesn't route Internet traffic through a LAN (except in very specific circumstances usually relevant to corporate networks with fixed Internet-accessible IP addresses on all machines). The freeware AnalogX proxy server might be sufficient to the task, depending upon what-all Internetting you want to do from your Axim.
I'm using a freeware app called
nPOP to send and receive email directly from my Axim (well, via USB ActiveSync). Install it, fire it up, define your accounts and servers, and away you go. You should be able to find the download via a web search.
(Dunno if you can do this with Outlook as well. I hate Outlook for email -- absolutely loathe it. <_<
'Hope this makes sense and helps,
-Bill, Ottawa, ON