Essentially, I boil water, throw in the dry ingredients, let them simmer for half an hour, and remove the wort from the heat. I've started adding the sugar at that point so I don't get sticky splatters all over the stove. Usually I only simmer the ingredients in half the water. Once it's cooled a little, I strain the wort and pour it into a gallon jug with the rest of the cool water. When it's down to 100 degrees (or the temp suggested on the yeast package), I pour a little into a dish and add the yeast to proof for 15 minutes and then pour that back into the wort. I've found that I get more evenly carbonated bottles if the yeast is totally dissolved (and thus evenly distributed.) I cap and shake the jug a bit to mix and aerate.
Then it goes into the bottles and capped. I use a powdered sanitizer to get everything clean, though I don't think it's as crucial as it is with beer, since it's only fermenting for a few days.
As far as equipment goes, you ought to be able to make do with what's in a regular kitchen and some used plastic soda bottles. (And I've had people highly recommend not using glass, since it will explode if you let it ferment too long with so much sugar.) I have managed to collect a few semi-specialized tools though.
I think this is all the equipment I use to make a batch:
pot for simmering
measuring cup and spoons
bowl for proofing yeast
strainer and finer splatter guard to strain
bottle caps and capper