Olfa Rotary Cutter - 45mm
I recommend Olfa rotary cutters. There are multiple styles of handles, and the Olfa Splash model comes in different colors, if that's your thing. (Yes, that's my thing. Santa, I want the purple one for Christmas!)
Fiskars 18x24 Inch Self Healing Rotary Cutting Mat
This is the mat that I currently own. I have also used Olfa mats, and those are high quality, too. Make sure it says "self-healing" and that it's double sided. Don't let your children write all over yours with a pencil. I need to clean that off :)
Omnigrid 6-Inch-by-24-Inch Quilter's Ruler
Omnigrid 6-Inch by 12-Inch Quilter's Ruler
I have had great experiences with Omnigrid rulers. They also have a line of rulers called Omnigrip, that are supposed to grip the fabric better for less slippage when cutting. The other ruler shown is a Creative Grids brand, which I also love. If you end up getting another brand, just make sure that it has lines that run the length of the ruler every 1/4".
Straight quilting pins
Here's an example of what these look like.
Dritz Quilting Curved Basting Pins Bonus Pack, Size 1, 300 Count
I have no brand preference for these, but try to get size 1 pins.
Water Soluble Marking Pen, Blue
Again, I have no brand preference here. I just wanted you to see an example of what one looks like! This product will not show up well on dark blue, black, or purple fabrics. If you are using those colors, you will need to get tailor's chalk or a chalk pencil instead.
Hand Sewing needles and sewing machine needles
You will need them both. It's very important to start a project with a new sewing machine needle.
It's embarrassing how much I actually use my seam ripper. The large one I linked to is much more ergonomic that the really small ones. I don't know why I'm still using a small one!
I love using starch on my fabric before cutting it out. It makes the fabric nice a crisp and it makes it so much easier to cut and sew accurately. I love Best Press and Faultless Heavy Starch. Best Press is more expensive but comes in lots of lovely scents. Faultless is cheaper and available at the grocery store, so that's what I use most often.
A walking foot is used for straight line quilting. It will give you much more even stitches than you would get by quilting with your regular piecing foot. It's very important that you get one that fits your machine! The one I have linked to is a generic brand that works with Singer, Brother, and Kenmore sewing machines. To get one guaranteed to work machine, search for (your machine brand) walking foot.
Quarter Inch Presser Foot
This isn't absolutely necessary, but it's so helpful. Again, like with the walking foot, make sure you find one that fits your sewing machine.
permanent fabric marking pen
I use these pens on my quilt labels. Always label your quilts, people. Always!
As far as a sewing machine.... I say start with what you've got. I started with a hand-me-down sewing machine that was probably only $100 when it was new. And I made 5 or 6 beautiful quilts with it before I decided I wanted something else. (I currently sew on a Bernina 1630). Before you go out and purchase something new, see if anyone if your family has something sitting on a shelf they aren't using anymore. For machine piecing and quilting, all you need is a good quality straight stitch and that tends to be where older machines shine! Using what you have for now will give you time to learn what you really want out of a machine... and time to save up for one that meets your needs and your budget.
I hope that was helpful! New quilters, I just know you're going to love this wonderful hobby. Experienced quilters... did I leave anything out?