Hi TheMud!
Although the A1200 is slightly better and more expandable, I think the A600 is still one of the best Amigas to have, maybe it's the second easiest to upgrade these days thanks to the ACA630, A604 and the ease of use of the PCMCIA port and standard 44pin IDE controller.
I do recommend getting some PCMCIA cards, like a CF or SD card reader. The one you linked to will be fine, but remember these parts are quite common and many people are trying to take advantage of Amiga users by selling these parts at over double the regular cost, so shop around. The driver software is available for free on Aminet, and I'm sure it wouldn't be too much effort for someone to send you the drivers on a disk. In fact I'm starting a global disk swapping network for this very purpose.
A PCMCIA ethernet or wireless card will also be very handy, as even a stock A600 can go online with one.
If you don't have a 1MB ChipRAM expansion already, you can't get anything better than the brand new A604, which is available at all the online Amiga retailers. Not only will it expand your A600 to a total of 2MB ChipRAM, but it will provide you with a battery-backed real-time clock, an adapter to plug in an Indivision ECS, and TWO clockports! One is a standard clockport, suitable for all A1200 clockport expansions like high-speed serial/parallel cards, sound cards, or the Subway USB card, which will perform almost twice as fast if it's used on the new high speed clockport which is also on the A604.
If you don't want to shell out for the ACA630 accelerator, which I HIGHLY recommend, you could get a simple PCMCIA SRAM card to bump up your FastRAM, and get a slight speed boost as well. 1MB, 2MB and 4MB cards are available, but they will cost you almost as much as an ACA630, and that has a 25Mhz 68030 and 32MB RAM on it. Also, if you have a SRAM card taking up the PCMCIA slot you can't go online unless you use a serial cable connected to a PC (for dialup-speed internet access, which isn't bad for IRC, Email, MUDs and stuff anyway).
I'm lucky enough to have two A600s, so I have a system set up both ways, and they both have their charm. One A600 is a powerhouse that can run the latest OS3 software, while the other is perfect for those older, simpler applications and great for classic games.
I have a link to a 4MB SRAM card for sale if you'd like to purchase one, I won't post it here because it'll very likely be gone by the time you see my post, there are too many lurkers here ready to snap things up cheap and resell them for profit. It's 85 Euro, where as you could get an ACA630 for around 160 Euro.
For internal storage you can't go past a Compact Flash adapter and a 4GB-8GB Sandisk CF card. The 4GB ones will work perfectly without requiring any patches, as long as you create your partitions smaller than 2GB, so just make two 2GB partitions if you get a 4GB drive, and you're set. For a 8GB or larger drive you'll need some patches and better filesystems, which can be a little tricky to set up at first.
As has been suggested, it's best to use WHDLoad and play games from the HDD/CF these days, it's much faster and trouble-free, especially when the data retention of HD media is bad for DD formatted disks.
Definitely get Kickstart 3.1. The first thing I do with my Amigas is upgrade them to 3.1, or at least stick in a ROM Switcher with 1.3 and 3.1 on it, since 1.3 is the best for running some old floppy based games.
Also, if you haven't got any good game controllers for it yet, these ones are only $4.55 each with free shipping, and are perfect for Amiga games -
http://www.dealextreme.com/p/additional-gamepad-controller-for-sku-28970-30530