My question is, where to from here.
What is the cheapest, easiest system that I could put together to get the cards up and running as soon as possible?
I realise that I'll need a hefty power supply, and I'd like to leave a little room for expansion, so if possible a mobo with a third slot would be good. Although if it's gunna be expensive, then two slots would be fine for now.
Cheers
You mention you want some room to expand. If that's the case then think about what you might want to add, or if at some point you're going to change out that 5830 for something with a little more bang. If you think you'll work your way up to 3 5970's, get a ~1200watt PSU. If you think you're going to stick with the one 5970 and fill the other slot with some other single GPU card, I'd suggest an 850w. Noteable point: Efficiency decreases as you near the rated output wattage, so ideally you'll want to have your total load be somewhere around 80% of the rated capacity. This is not a rule, but a best case scenario.
PS: Don't cheap out on the PSU Do some research as to who makes what and their quality rankings. (FWIW, I use XFX and am extremely happy with them)