Perhaps the single most important technological advance since the sewing machine, as far as quiltmakers are concerned, is the rotary cutter. This nifty little tool, which looks like a pizza cutter, allows you to cut fabric far more accurately and quickly than our grandmothers ever could with scissors.
Rotary cutters come in several sizes and many different configurations: I like the ones with ergonomic handles and especially like the ones where the blade automatically retracts when you put it down. I have done serious damage to myself with rotary cutters -- they are SHARP and you can easily draw blood by barely brushing against the blade, so you want to keep it shielded.
You need two more tools to go with your rotary cutter: a self-healing mat, and some kind of plastic straight edge. Buy the biggest mat you can afford (wait for the half-price sale). As you get more serious about your quilting, you may want to buy additional mats so you can line them up and trim the whole length of your quilt with one swoop of the blade. I also have a small mat that I keep next to my sewing machine, in case I want to trim a block or cut a new strip without getting up.
(I'm going to give a lot of instructions and tips about how to cut pieces with straight edges, as you need for traditional pieced designs and for many improvisational quilts. But you should know that you don't have to cut ruler-straight edges. It's possible, and very pleasant, to use the rotary cutter to make curves, or for what I call "freehand straight lines," those that may have a slight wobble. You can buy plastic templates to cut exact quarter-circles for blocks such as Drunkard's Path, or you can cut curves freehand.
(I recommend you read the rest of this post about ruler-cut seams even if you plan to work without a ruler; you may pick up some helpful hints. Then, to read about curved cuts and curved seams, go here and here and here.)
Like cutting mats, plastic straight edges come in many different sizes and shapes. I have two long rulers as well as several squares, handy for trimming blocks to size. It's no accident that many of my quilts have blocks that finish to 5 1/2 inches square -- that's what you end up with when you use a 6-inch square to cut the blocks in the first place.
To simply cut a straight edge with your rotary cutter, spread your fabric out smoothly on the cutting mat. If the fabric is wrinkled, press it first; otherwise your cut may not be straight and your subsequent construction may be sloppy. If you're right-handed, hold the ruler firmly in place with your left hand and cut along the right edge of the ruler.
Some people like to cut away from themselves, others like to cut toward themselves. I strongly recommend cutting away, simply from the standpoint of safety. I don't want that blade going into my clothes or my stomach. Some people like to hold the ruler pointing away from them, others like to hold it left to right. I'll assume for the rest of this discussion that you will cut away from yourself, at least on the long cuts.
Much of the cutting you will do in quiltmaking requires some measurement or alignment, and that's where the grids printed on your mat and your plastic rulers and squares come in. (If you're cutting freehand, or cutting straight edges but don't need an exact size, you may find the grid on your mat distracting. If so, flip the mat over and use the unprinted back side; it performs exactly the same as the front.)
Sometimes you want strips to be a certain width, other times you don't care exactly how wide they are but you do want the edges to be parallel. There are two ways you can proceed: one is to work with the grid on the ruler, the other is to work with the grid on the mat.
In either case, here's how you get started.
Press your fabric. You'll probably want to cut strips crosswise to the grain (that is, your cuts will go from selvage to selvage). The fabric comes off the bolt with a fold down the center, but the cut or torn edge may not be exactly square to the fold. So arrange the fabric carefully on your board with the fold exactly aligned with one of the grid marks. The selvages should also be parallel to the grid.
Align your ruler with one of the grid marks perpendicular to the fold, and slice straight across. Check your work by opening up the fold and making sure the cut is straight (no angle where it crosses the fold). Now you have a straight baseline edge to guide all your subsequent cuts.
To use the grid on the ruler, place the fabric on the cutting mat with the baseline edge at your LEFT (if you're a rightie). Decide how wide you want your strip to be, and find the line on your ruler that far to the left of your cutting edge. If you're going to cut several strips the same width, it's smart to mark that line with blue painter's tape or with a grease pencil (it rubs off the plastic when you're done) rather than having to figure out each time which line you want.
Position that line on the baseline cut edge of the fabric. Check to make sure the edge of the fabric matches the line all the way up and hasn't slithered out of alignment somewhere. Now slice across the fabric.
If you're careful when removing the strip you've just cut, and it hasn't pulled the big piece of fabric out of line, you can make your next cut by just moving the ruler to the right, but always check to make sure the edge is aligned with your gridline before you cut.
To use the grid on the mat, place the fabric on the cutting mat with the baseline cut edge at your RIGHT. Align the baseline edge with a grid mark on the mat. Decide how wide you want your strip to be, and find the line on the mat that far to the left of the baseline edge of the fabric. Position your ruler along that gridline and slice.
Again, if you want to cut a lot of strips the same width, you can mark the lines with painter's tape (not grease pencil; it won't rub off the cutting mat the way it does off the smooth plastic rulers).
The major difference between the two approaches is that when you use the grid on the ruler, your ruler is covering the strip. When you use the grid on the mat, your ruler is covering the rest of the fabric. With wide strips, there's no practical difference between the two approaches. But when you're cutting narrow strips, I find that simply holding down an inch or less of fabric doesn't give enough stability to the setup. It's too easy for the weight of the fabric to pull that inch of fabric out of alignment, of for your hand to brush against the bulk of the fabric and jerk it out of position, or even for your rotary cutter blade to catch on a seam or slub and pull the fabric to the side. So the narrower the strip, the more likely I am to use the grid on the mat.
I find this cut precarious -- there isn't enough fabric for the ruler to get a grip and hold things in place while you cut
If you find fabric slipping worrisome, you can buy little cork stick-on feet to apply to the corners of your ruler or square. That way the fabric will never slip while you cut.