Putting the new harddrive in the old system is the easiest way.
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The LAN option is the one i would use, because it's considerably faster than a null-cable (think modem connection). It's also a convinient way to share files. And, of course, share an internet connection.
You can use a cross-over cable, if you are only connecting 2 PCs, or connect the PCs through a hub (or switch). The first is inexpensive, the latter is... not quite inexpensive, but you can connect more than 2 PCs for you home-LAN. Great for Multi-Player.
For the first you will need:
2 NICs (10/100 Mbps, Full Duplex** if possible)
1 Cross-over Category 5 UTP cable*
And the second:
2 NICs (10/100 Mbps, Full Duplex** if possible)
1 hub (10/100 Mbps), or switch (10/100 Mbps, Full-Duplex**)
2 Straight-Thru Cat. 5 UTP cable
* See Kazan's post if you have to make it yourself.
**Full-Duplex means transmitting and receiving at the same time.