A cutting mat, a rotary cutter, a ruler, scissors and shears, template plastic, Sharpie markers or Pigma, quilt batting, basting spray, needles and thread.
Quilting takes cutting fabric for piecing or applique, sewing it back together, sandwiching pieced top with batting and backing, quilting and finishing borders. Here's basic list of supplies for every stage.
Self-healing mat, also called self-sealing or rotary-cutting mat: 24"x36" mat with alignment grid on one side, usually grey or green, which sustains repeated cuts with a sharp blade. It's washable, not slippery and good for laying out fabric, drafting, drawing and framing as well. It is not intended for pressing on. I did press small pieced blocks on the self-sealing mat with four times folded flannel on top of it. Do it on your own risk. Smaller mats, pink and blue mats, and mats with metric grids are available as well. X-Acto, Olfa, Omnimat and Scotch are most common brands, a large mat costs up to $70, but they are often on discount elsewhere.
Cutting fabric: takes precision. You will need large tailor's shears and fine-point manicure or embroidery scissors. I use Fiskars I bought in Target. I replaced my tailor's shears after five years of moderate use and didn't bother with presumably possible sharpening. You will need a rotary cutter for strip-piecing, maybe several cutters with different blade diameter. Olfa is a default brand for rotary cutters, but other brands are available in the same art and craft supply stores. The 60 mm cutter costs $18-$30 and more, between the sale price for the basic cutter and the list price for more ergonomic models.
You will need a ruler for strip-piecing and for planning and layout. Best rulers are long, wide clear plastic rulers with 1/4" grid and a lip to hold onto the edge of the mat. O'lipfa, Omnigrid and Quickline are some of the names. I don't know of any difference between the brands. Templates: art and craft supply stores sell template plastic sheets. I make my own templates recycling spare plastic from packages. Sharpie markers don't smudge too badly on plastic. I use transfer paper for applique templates.
Quilt batting: synthetic or cotton. Both are sold pre-cut to standard sizes or bulk, by yard. Standard sizes are Crib, Tween, Queen and King. Bulk batting is always cheaper. Synthetic (polyester, Poly) batting is loftier, it s hard to quilt as fine as cotton and it tends to escape through the top and show on dark fabrics. Due to its bulk, Poly batting is harder to align and baste with the top and backing. It will not fit under some walking foots in sewing machine. Quilts with Poly batting look and feel more comfortable, and free-hand machine quilting or machine trapunto looks more dramatic if you manage to do it. Cotton batting is thinner, firmer, more traditional and more manageable. I use unbleached cotton batting for my storytelling quilts and sometimes add a layer of Poly batting when I need more relief. I use unbleached out of unfounded superstitious idea that bleach residue can somehow destroy fabric I use.
Basting: use safety pins (not straight pins), basting stitches or basting spray. Quilt Basting Spray from Sullivans USA is too sticky, it smells chemical weapon, it doesn't wash off completely, it soaks through the fabric, makes your quilt stick to the walking foot, and gums the needle. Still, it works. It eliminates basting and allows hand-quiltng with no frame. I bought it to replace the can of Dritz Repositionable Spray Adhesive and still use it, but next time I'll buy something different, like June Tailor Quilt Basting Spray.
Sewing machine -- any sewing machine that stitches straight is OK for quilting. Cottons don't fray, don't slide or slip and are easy to work with.
Thread: cotton coated Poly for machine piecing and quilting, hand-quilting cotton thread, embroidery thread for details. Deep navy and off-white threads are most used. Invisible thread, or monofilament comes in clear (light) and smoke (dark), it creates interesting effects, but turns brittle with time. You will also need a darning foot, hand- and machine-needles, long straight pins with large heads, and safety pins in different sizes.
See also: sewing machine, quilting fabric, picture framing.