I heartily recommend a TCP/IP connection and AmigaExplorer.
The A600 can use them.
Once you have 2mb chip (trapdoor expansion) you will be able to use a PCMCIA ethernet card (you can get them for about £10), make sure it's either an NE2000 compatible one, or the 3Com 3c589 one.
You need 2mb to run the TCP/IP stack such as Miami or AmiTCP.
A hard drive is an easy upgrade, any old laptop 3.5" IDE drive will do, almost any size as Amiga programs are small. You can usually get 10gb drives for under £25.
This is a good website
Take a look at the guide I wrote for A600 users