Wish I'd bought mine into work now :)
Hard wired should work with no problems, and wireless should be pretty straight forward.
OK, from memory, you should have a little icon that looks like a computer near the clock, if you right click on this it should give you a couple of options two of which are to enable networking and enable wireless, make sure both are selected.
Once thats done you should be able to click on the network monitor icon next to it (looks like two screens when not connected) and it will bring up a list of wireless networks, just select yours and enter the key.
If you are not getting the options can you remind me of the model of eee it is and what version of eeebuntu you downloaded.