If you are making pot holders that need to withstand heat of hot dishes, you would need to use 100% cotton batting and possibly choose one of the heat-resistant fabrics available at a full service fabric store.
When purchasing your 100% cotton batting, be sure to check the label. Some brands of batting claim to be 100% cotton but use a polypropylene scrim -- this batting would not be a good choice for potholders. Read the fine print of the package carefully and if purchasing from the roll, check the end of the roll.
Heat-resistant fabric is a heavy weight closely woven fabric that has a shiny silver finish on it. It is sometimes used for ironing board covers. Many of the commercial potholders put decorative fabric on one side and use the heat-resistant fabric on the reverse side. I have also seen this fabric used as an inside lining for curling iron cases or lining a bag to place your hot travel iron.
For placemats or table runners I would again suggest a cotton batting. A thin batting would be a good choice as you don't want your dishes to rock back and forth on the placemat. If you will be placing hot items on your placemats you would want to be sure your batting is 100% cotton.
I also got the following suggestion from subscriber, Diane: I use old baby receiving blankets instead of batting for my placemats -- just thick enough and they do not shrink -- usually very inexpensive at garage sales.
Also, I would like to publish an answer to one (very broad) question because I think many people would benefit from this information.
For pot holders I use flannel on either side of a piece of old washable wool from a skirt or trousers, then a pretty top and back. This is heat proof and flexible.
I'm just trying to get started, and I need information on the tools needed and the best economical quiting frame to buy.
For hand quilting I would suggest a 14 inch quilting hoop on a stand to put in your lap. When hand quilting you need to have one hand operating the needle and the other hand under the quilt. You will find the stand helps with ease of quilting.
Beginners should start with size 10 between needles. The needles seem small but a smaller needle will help you make smaller stitches. Strive for even stitches at first and as you practice your stitches will get smaller.
You will want a good thimble that fits the middle finger of your quilting. There are ergonomic thimbles available and I recommend you check them out. If you quilt many hours you might appreciate this type of thimble.
Use Quilting Thread for hand quilting. Many brands have a special finish for ease of quilting. If your thread twists and breaks you might want to use a thread treatment or beeswax.
A dear friend suggested using the tips of rubber gloves (playtex) for a thimble, as I have a problem using the thimble finger, and the glove tips fit perfectly to my finger, I also use the one for the thumb.
For machine piecing, your largest investment is your sewing machine. Buy as good a quality as you can afford. It might be better to buy a 'pre-owned' good quality machine than an inexpensive new one. Features that are useful for a quilter - lowering the feed dogs (the teeth that pull the fabric through the needle), availability of a 1/4" piecing foot, a walking (even-feed) foot, and a darning foot.
Good quality machine needles are important. Buy the brand your machine manufacturer recommends for your machine. I use size 11 for piecing and size 14 for machine quilting.
Use quality thread. Many quilters recommend 100% cotton thread. Some use cotton covered polyester. Both work well. Please do not use the 'discount' thread. Your quilt is worth the small extra expense of good quality thread.
Rotary cutter, mat, and ruler. Accurate cutting is essential to accurate machine piecing. Your first mat should be at least 18" x24". Buy a larger mat if you have a larger space to use it. You can always buy a smaller one later to carry to classes. A 45 mm cutter is a good size to buy for your first one. There is one larger and two smaller. Ask at your quilt store if you can try different brands and see which size feels good in your hand. Some of them are ergonomic with a curved handle. These can be set up for either right hand or left hand. I prefer the straight handle cutter as I cut with both hands. A 6" x 24" ruler is a good size for a basic all around ruler. You can add a 6" x 12" and square rulers later.
For piecing at the machine you'll need some small scissors for trimming threads, a 1/4" piecing foot, and a good seam ripper. Those little seam rippers that come with your machine are just too small for regular use. Also you'll want some pins. I like the long 'flower' head pins available at the quilt store.
Most of these tools are a long term investment. If you buy good quality tools they will last a long time.