What files do I need to copy to a new computer to prevent having to sync with the network?
https://en.bitcoin.it/wiki/Data_directory

This might be of some help.I followed that and successfully moved the files from C:\Users[username]\AppData\Roaming\Bitcoin\ to E:\BitcoinData.

Just cut and paste the whole content of from the Bitcoin folder (C:\Users[username]\AppData\Roaming\Bitcoin) and set the new target as 

"C:\Program Files (x86)\Bitcoin\bitcoin-qt.exe" -datadir=d:\BitCoinData and that is assuming you have bitcoin Qt installed in your C: drive.Change the path accordingly.