This isn't the only way to do things, but it's how I've been printing.
First I draw the design. If it's simple I work straight onto the linoleum, but if it's fairly symmetrical it's useful to draw it on tracing paper so you can check that both sides are the same. If it's a complex design you'll probably want to work it out on paper first anyway, then trace it to get clean lines to work with. If it's something that reads better in a particular direction (like lettering) it will need to be reversed, as the print will be in reverse.
Next I trace it onto the lino block using carbon paper, and then go over the lines with ink. If you have a really steady hand you could probably skip the ink stage, but it's useful to make sure your lines are accurate, and to prevent smudging while you're cutting.
For fabric printing I've found that thicker lino is easier to print with; about 5mm thick is great.
For some reason I prefer to use woodcut tools for cutting lino, and I find a V-gouge the most useful. I think they're easier to sharpen than lino tools; but, depending on the kind of mark you want to make, you could use anything from an ice pick to a craft knife.
I cut all the outlines first, and then clear out the bigger areas using U-gouges of different sizes. For fabric printing you'll need to cut deeper than for paper printing, because the lino block is pressed onto the print surface, rather than the other way around. I tend to cut almost down to the backing (another reason thicker lino is better).
The thick lino is often harder than the thinner kind. It's important to keep lino warm when cutting, so that it stays fairly soft. This minimizes the risk of the tools slipping and ruining your block or your fingers. In summer it's not too much of a problem, but in winter you'll need to warm the lino every now and again as you work on it. The simplest way to do this is to work on two blocks at once: sit on the one you're not working on. As soon as the working block starts getting cold, swap it for the one you're sitting on.
*Update: I no longer use traditional linoleum, rather a plastic one. Traditional linoleum doesn't stand up to repeated printing with water-based ink.