Here is a link to a video for how to do a long tail cast on by what is called the thumb method. I learned this way years ago and continue to use this method most of the time. It makes exactly the same cast on as shown on this site on their long tail video, but I think it is easier to learn. And I like it.
It begins with a slip knot on the needle. The gal on the video doesn't go into how to do the slip knot much. She indicated that she had shown that before, on another video I guess. She does one a pretty slick way, but any way you know or can learn to do a slip knot on the needle to get started will work. I don't do mine the way she does.
The text above the video and the gal says this makes a "tight" cast on. I wouldn't say that necessarily. It is the most common, standard cast on and will work in most circumstances. Don't do it too tightly, in other words when you pull the yarn in your left hand to tighten the stitch don't pull it too hard. You want the resulting edge to stay even too and you can see how it is coming with each stitch and you can take out part of them (or all) and do them over if you want.
If you find that as you knit your first little swatches that the cast on seems to pull in a little you may be doing it too tightly. One thing I do commonly is to cast on on a bigger needle (or hold 2 needles next to each other as one to cast on--I find this awkward though) and then start knitting with the smaller needle. But many times you don't need to do that, just don't pull the stitches too tightly as you go.
LINK TO VIDEO