A basted quilt top or small project (maybe a pillow top to start?)
Hand quilting thread (100% cotton, usually waxed or coated/glazed. I prefer YLI.)
Quilting needles (Betweens are the needle of choice for quilting, size 8-12)
Small thread snips
Quilting hoop (this is like a monster embroidery hoop)
Of course, you can quilt with regular needles and no hoop, but I have found it makes it much easier to have the "right" tools. There is also perle cotton quilting which uses a type of thread called perle cotton, which is thick like embroidery floss, and larger-eyed needles. This gives a much showier, larger stitch and really stands out on modern quilts. A good example is here on Molly Flanders Makerie.
The most common quilt stitches are the stab stitch and the rocker stitch. Stab stitching is where you stab the needle straight down and then straight back up along your line of sewing, making one stitch at a time. Slow going, but necessary where there are a lot of layers or tricky bits of a design (like tight curves). The rocker stitch is what is considered the "quilting stitch" by most people and is what I am going to attempt to describe. If you are reading this you are probably a quilter already so I will not describe quilting designs or marking your quilt.
Put your quilt sandwich into the hoop and snug it down fairly tightly. It should have some give as you need to be able to push up on it from the bottom. Decide where you want your line of stitches to be and insert your knotted thread about a half inch from the right-hand end of the line (reverse if left-handed), going just under the quilt top and batting. Bring the needle up where you would start your stitching and pull the knot into the batting to hide it. With your left hand under the quilt hoop, feeling for the tip of the needle, insert your needle straight down just until it comes through. Then "rock" the needle down parallel to the quilt top with your hand in a C shape while pushing down with your thumb. Simultaneously push up with your middle finger from the bottom to create a small ridge or hill which the needle should go through. (Just until you feel/see it.)
That is one stitch! Now while keeping that stitch on the needle, bring it straight up and down again and do another. I usually do 4 or 5 before I pull the needle, but my grandma would get 10 tiny, perfect ones on there each time. I use a small spring loaded pliers to grab the needle and pull it through as it is easier on my hands, but that is just me.
Here is a good one from Sew, Mama, Sew! by Sarah from Hip to Piece Squares. If I need inspiration for designs, I usually search for sashiko designs or just follow the seam lines of my quilt blocks. While it is definitely more time consuming to hand quilt, it has its own way of being rewarding. Give it a shot and don't forget to "Take it With You" this summer!