Cutting flagstone isn't as difficult as it looks, depending on what kind of cut you need. For outdoor patios, landscaping and other common uses, you'll frequently need an approximate cut rather than an exact one. In that case, it's often a matter of creating a controlled break in the stone rather than a precision cut. When you do you need exact cuts--say, for a set of steps--talk to your local home improvement store about renting a masonry saw.
Hammer and Chisel
For cuts that have to be approximately (but not precisely) straight in relatively thin flagstone, you can break it along a scored line. Set the stone on a solid setting. Mark your line on the surface in chalk. Use a hammer and heavy iron chisel to chisel the line into the stone. Once a groove has been cut, continue tapping at it with the chisel and hammer, moving it back and forth along the length of the line. Don't hit it hard; the point is to chip at it gradually until the stone under the line weakens enough to break.
For approximate cuts in flagstone that's too thick to chisel apart, buy a masonry bit for your circular saw. You won't be able to cut all the way through the stone this way, but you don't have to; the point is to get a good, deep groove that can then be knocked off with a hammer. Set the blade shallow (1 inch or less), mark your line on the back of the stone, and cut very slowly along it. Once you've cut a groove along the whole line, set the stone on the edge of another stone, so the grooved line is just off the end, and hit the over side of the line hard with a hammer.
For exact cuts, buy or rent a masonry saw. This is similar in concept to a circular saw, but is much more powerful, and the blade is larger. Don't try to push the saw through the stone as you would cut a piece of wood, but rather set it in place, hold it steady, and let the power and weight of the saw do the work. The saw produces a lot of noise and dust, so wear goggles and ear protection.