Both piecing (assembling the quilt top) and quilting (sewing the quilt top to the backing fabric with batting in between) can be done without a machine. For the piecing, you'll need to be able to stitch with smallish stitches (to keep the pieces together), but after you do it for a while, this becomes quite easy. For the quilting, typically you just want to have even stitches (but if you want to get arty, anything is possible!).
It is possible to cut pieces with scissors, but a rotary cutter is much faster. But it's also more dangerous, since a rotary cutter is razor-sharp. If you're buying a rotary cutter, you'll need a decent-sized cutting mat (at least 24 inches in one dimension) and quilting straightedge (they look like clear plastic rulers; a 6"x18" is a good starting size). Rotary cutters really started a revolution in quilt-making, taking the drudgery out of cutting large amounts of fabric.
For hand sewing you'll need a standard needle, thimble, small scissors to snip threads, and 100% cotton hand-sewing and hand-quilting thread. Yes, you can use other thread -- but 100% cotton that's made for hand-use will be much easier to work with. Generally quilters use off-white, beige, or gray thread for piecing. You want the thread used for piecing to "disappear". It's not like when you make clothing and match the thread to the fabric. For quilting, it's entirely up to the quilter. Sometimes you want to see the thread, other times you want the thread to disappear and just let the puffiness of the batting show off the quilting pattern.
For batting, I'd recommend cotton batting. Polyester has a higher loft, but it can be harder to hand-quilt this material. Some poly-cotton blends are reportedly easier to hand-quilt. A widely-available standard batting is "Warm-and-Natural" or "Warm-and-White". It's a thin bat that gives pieces an "heirloom" quality.
I highly recommend a book for instructions, not simply a pattern. The patterns have basic instructions, but for a first quilt it's nice to have lots of details and options. Most beginner books have several projects in them. There are many books out there, but I think the key is to find one that has some projects you think your daughter would like. Lots of books have very "country" or "overly cute" quilts; if this isn't her thing, then make sure the book has at least one pattern that's more flexible.
I suggest starting with small projects; a small wall quilt, table runner, pieced pillowcases, etc. (Bigger projects can take so long to finish that a beginner runs out of steam.)
As for fabric, the amount you need depends on the pattern. Also, the configuration of the fabric depends on the pattern. In the quilting world, there's such a thing as a "fat quarter". This is a quarter yard of fabric, but it isn't cut selvage-to-selvage. It's approximately 18"x22"; half a yard long, and half the distance from selvage-to-selvage (approximately 22"). This size is popular with quilters who like to use lots of different fabrics; they get a small amount of fabric but you aren't limited to 9-inch-wide pieces (the standard quarter-yard would measure 9"x44"). Some patterns require fat-quarters. Some patterns require "skinny" (standard) quarters. It all depends on the pattern and how it's put together.
Of course, you can avoid this buy buying fabric in half- or whole-yard lengths. That's more expensive, but a few nice pieces of fabric for inspiration are a good start.
I recommend buying quilter's (cotton) fabric. It's more expensive, but it's much higher quality. That means it is easier to work with (unless you find some handmade loose-weave stuff) and it has a really nice "feel" to it. Cheap cotton "calico" fabric runs about $3-$4 a yard; quilter's fabric is about $8 a yard. Quilter's fabric typically has better dyes and a better print job, too. More and more you can buy quilter's fabric at regular fabric stores -- go by price. Stick with cotton, too -- it's much easier to work with and there's lots of gorgeous colors and prints out there. (Btw, "quilter's fabric" really isn't a term; I'm just talking about high-quality woven cotton fabric meant for use by quilters.)
For inspiration, you might want to pick up a few quilting magazines. They're always great fun to look at. Most of them are quite "traditional," but your daughter's tastes are more modern, I recommend "Quilting Arts" (http://quiltingarts.com/qamag/qahome.html
Hope that helps.