Welcome A600 user!
All things leirbag28 said was completely true.
If you want a HD bigger than 4GB you must have Kickstart 3.1 ROM, although you can use up to 4GB HD with Kickstart 37.350 ROM as well.
I recommend as leirbag28 did to have a look at
www.amiga600.de (by Christian Krenner)
www.amiga600.net (by Mike Parker)
These are A600 fanatics that made a great upgrade to their miggys!
I inspired by those guys and I have an upgraded A600 with
-68030@42Mhz+68882@40Mhz FPU
-HD 3.2GB (I don't need a larger one!)
-4MB Fast RAM, 2MB Chip RAM
-Compact Flash to IDE Adapter (with 256 CF Card)
-PCMCIA Ethernet NIC
-3.1 Kickstart ROM
running in AmigaOS 3.9 BB2
Feel free to ask anything you want!
As for games you can download WHDLoad from
www.whdload.deand download many many games to use on your HD without using floppies!