First up - welcome!
Now - do you want to make large quilts or small quilts? This might make a difference to the quiltmaking method you want to learn first. For instance, if you start by making a small quilt you can try one method - perhaps hand piecing. If you don't like that method or you feel it would be too much work to make a large quilt that way you might want to try another method.
In the past quilters have made magnificent quilts with just needles and thread, scissors and fabric. Some quilts can even be made with no cutting at all. Manx quilts are made with torn strips, and whole piece quilts are large single pieces of fabric with intricate quilting patterns.
You can make quilts on your sewing machine or by hand or by a combination of the two. Do not dismiss hand piecing a quilt. It is surprisingly fast once you get going, and very portable so you can take it with you and snatch 5-10 minutes here and there to make blocks instead of having to wait until you can sit in front of your sewing machine.
The mat and cutter your referred to are for rotary cutting. This is how many quilts are made now. The rotary cutter looks like a pizza wheel, and when used with a quilting ruler it gives very accurate cutting of straight lines. The rulers (special perspex quilting rulers) are essential to go with the cutter, and they give accurate measurements for your cutting. The mat is what you cut your fabric on when you are using a rotary cutter. The mat is made of a special material that will protect your table or bench but will not be damaged by the very sharp blade in the cutter.
If you are going to start with rotary cutting for your quilt making you will need a mat, a 24" long quilter's ruler, and a rotary cutter as a start. This can be a significant investment so if you are not absolutely sure quilting is for you consider making a quilt using another method of cutting - good old-fashioned scissors. If you are really sure this is for you it is worth buying good quality tools right from the start.
As with most crafts, quilt making is probably best learned by seeing. If you don't have a local quilt shop (LQS) that can help, check around for a local quilt guild or group that might have classes. Or ask if any friends know a quilter who might be willing to show you the basics. Many quilters who do not teach will demonstrate the basics of cutting and pressing for a beginner.
I teach quiltmaking for beginners from my home and I encourage everyone to make lots of small practise blocks and sew them together into mini practise quilts to learn the basics before they start cutting up "good" fabrics. My beginners have classes once a week and they don't even choose the fabrics for their first quilts until about week four or five, but this is clearly not possible in a situation where you are attending classes in a shop. But if your are teaching yourself, buy some cheap fabric with start with simple practise blocks. They make great palcemats and animal blankets so they needn't be wasted either.
For a first quilt I recommend something like a four or nine patch design if you are working on your own. If you can find someone to help you over the basics you might like to try something a bit more advanced like a Round the World or simple Bargello. These are quilt patterns that are usually made with rotary cutting methods. If you can find a class to attend they will most likely offer a sampler design or one of the ones I mentioned above (four patch, nine patch, etc).
It is entirely possible to learn quilt making from books. I know because I did it 30 years ago when there were few books and almost no shops for quilters here in Australia. But it is easier and quicker if you can get some basic instruction right at the start.
If you have more questions just ask, but take it slow and steady and don't try to rush learning the basics. It is worth the effort to get them right.
How about telling us a bit about yourself? Where are you from (just generally - not specifics)? Do you have any QIs (Quilt Inspectors are usually four legged furries but any pet can be a QI - and they will consider your quilts as their personal property!)?
And of course the most important questions - your coffee and chocolate preferences. I am the odd exception in this group - I don't like either lol