From today's perspective, A600 is... annoying, which I've found out a few years ago. The architecture choices are hard to work with: the PCMCIA slot limits the amount of memory you can work with (well, unless you don't want to use it but it's almost indispensable, esp. for data transfer), the internal bus only accepts "slow" RAM - the desired Fast RAM (CPU exclusive) can only be added by expansions put directly over the CPU itself. Sure, there are Vampires and PiStorms but the compatibility of both can be a bit spotty, and if you're dumping that much on the system you must love it specifically, lest you'd just go with an A1200 which is superior in every respect.
On that note, currently the easiest no-brainer Amiga to actually have fun with without eating your budget too much is an A1200. Slap an 8MB fast ram expansion, an IDE CF controller, and 3.2.2 OS/ROM it's ready to go. WHDLoad installs exist for every important game and many obscure ones too, data transfer can go either via PCMCIA>CF card, or a Parallel Port contraption that reads microSD cards. And if you want more power (though I can't imagine why, unless you want to run Doom on it), TerribleFire cards aren't too pricey. 128MB RAM + 030/060 CPU is a substantial boost (note: games, especially early 3D, will break with that much power).
Just watch for leaky capacitors (thankfully there's a whole industry for fixing this), and for the love of god don't use the stock A1200 PSU, it'll kill your system (thanks Commodore).