What would you suggest as the best (easiest) way to implement the network communication part of a simple two player, turn-based game in Delphi? (I am not a professional!)

since its not a real time FPS, id suggest using TCP, its a bit easier to use than UDP because you dont have to check for losses and the like...

The easist way of implementing a compelete UDP or TCP network system is to use a number of delphi components that are around.

I personally recomend ''Indy'' or Internet Direct
http://www.nevrona.com/Indy/
Uses Blocking sockets, but its design is very good.
Also has a large number of demos included.

If you dont like blocking sockets (but I would have a look at the Indy package, as it is the official internet package or Delphi 6)
there is ''ICS and MidWare - OverByte Home Page''
http://www.rtfm.be/fpiette/indexuk.htm
Uses non-Blocking sockets.

Both of these are completly event driven.