Ouch!!!!
You're right, there's absolutely no reason for a new graphic card for office software besides the old giving up!
Maybe your dad should ask an additional one for a price before buying next time!
About the network:
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Connecting two computers is quite simple, especially if one pc runs XP and the other at least win98se.
First check the two computer if both already have pci network cards installed. If they have you can use them and the cable. If your dad still has a 10MBit connection (I think so if it's a couple of years old) it should have a round BNC connector that stands out at the back.
Otherwise you just need 2 10/100MBit pci network cards and a RJ45 (also called cat5) datacable. To connect them directly you need a so called crossover cable because 2 wires have to be crossed for a direct connection.
This allows you to connect 10MBit and 100MBit cards if needed.
Besides the crossover cable you can also use a external 10/100MBit switch and connect both computers with normal RJ45 (Cat5) cables to the switch (there no additional installation needed besides the network cards in the computers).
I'm not sure about the prices in $ but at the moment I'd pay about 10 Euro for one network card and around 35 Euro for a switch with 5 connectors (for 5 computers). With the Euro course a bit higher than the $ I think it's about the same in $ as most of those parts come from the states!
After installing the cards and connecting them you can run the Homenetwork setup program in XP and you can use the add-ons on the XP cd to run the same on a win98se (not sure about win98) or winME system.
That alone should work but else go into the control panel, network and search for the TCP/IP protocol of the network card. There you can set the IP address. On the server the IP is "192.168.0.1" and on the other computer "192.168.0.2" (just raise the last number for additional computers). On both set the subnetmask to "255.255.255.0" and they both need the same workgroup (MSHOME should be standard for XP).
You also have to go to the "windows explorer" and allow access to the resources you want to share (there is a quite good help in XP to that!).
Changing computers:
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You need to do a new installation (never update windows to a newer version - I tried a couple of time and it NEVER really worked!) on the new computer (if it's not preinstalled).
To copy the data the easiest way is to install the old drive into the new one and copy all files to the new drive (make sure everything is good before deleting the original or even better make a backup before changing everything!).
Then you can remove it and put it back into the old system.
Celeron 500:
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Do yourself and your dad a favor and install at least 128MB (better 256MB) into the Celeron. I'm sure you're surprised about how much faster it works!!!!!
Just upgraded a neighbour computer with 512MB PC133 SD Ram instead of his old 128MB PC100 Ram and even I was surprised on how much faster it worked. In that case a new graphic card made sense because he also plays games on that computer.
With the 64MB TI4200 card (no reason for a costly directx9 card on an old 800MHz Pentium3 - even so his 3DMark scores nearly trippled) and the 512MB Ram together he paied less than your dad for the graphic card!