Since you are already networked with wires, there are at least two ways to connect your Axim. You can connect your Axim to your network with either a cable or using wireless.
1) If you want to connect the Axim with a network cable, you can get an ethernet CF (CompactFlash) card, something like
this..
2) If you prefer not having to be chained to the wall with a cable, then you will need a wireless (WIFI/802.11b) CF card for your Axim. A good number of the forum participants here seem to use the Netgear, Linksys, and DLink cards. I would suggest getting one of those cards since there's a very good chance someone here will be able to help you should you have any questions about the CF card. Personally, I got a DLink DCF-660W card because it was cheap. :)
Now for the wireless connection between the Axim and your network, there are two ways that you do this (yes, more choices):
1) You can get a wireless access point and connect that to your network. The Axim will then connect wirelessly to the access point. This is called infrastructure mode.
2) You can get a wireless adapter for one of your computers and install it on one computer. You will also need to use an internet connection sharing program on that computer. If you use Windows XP, then you can use the built-in Microsoft Internet Connection Sharing program. This computer will have to be left turned on when you want to connect to the internet on the Axim. This is called ad hoc mode.
You should also look at other threads and check out the other ideas that have been presented. These are just my personal suggestions. Hope this helps you and wasn't too techie.