I wouldn't go with serial or parallel. They're slow and unreliable. As you have A600, it has PCMCIA port and you can use real ethernet card in it. For example certain 3com cards (check Aminet for 3com and CNET drivers and compatibility lists).
Crossover network cable is needed if you want to connect Amiga directly to PC with RJ-45 cable. But if you have router or switch, then normal straight-through cables are enough. It's easiest to setup if you already have own LAN with router.
You can configure ftp server on either of systems... or why not for both, then you can transfer files from the computer you happen to use at that moment. There's ftp servers and client for Amiga as well as for PC.
Amiga nees separate TCP/IP stack, but stock A600 is bit limited with memory. You might be best to use some older AmiTCP which doesn't need that much memory for GUI:s etc as newer ones need (like Genesis or Miami). Configuring might need more work, but if you're familiar with TCP/IP settings it shouldn't be too big problem.
If you don't need A600 connected to net all the time, but you need it only for occasional transfers, you migth use this
NetworkBootDisk as well. You just enter your static TCP/IP settings to it and boot from it. Then you have ftp server ready for use without any further set ups, among the other things. Just dump the files with FTP client from PC to any storage media on A600 or other way around.