Most of the time, I knit toe-up socks. I find the casting on easier, and, especially if I'm knitting with handspun (as I now do at least half the time!), I find it easier to fit -as-I-knit (I don't have to worry about how many stitches to cast on!). Also I never have to fret about having enough yarn, which is a huge bonus with my not-so-consistent yardages of handspun...
So let's review the different cast-on techniques:
1. turkish - this one is the easiest. Basically you wind the yarn (tightly) around both needles and start knitting. Couldn't be easier. You need to choke down on the tension for the very first row, but apart from that, this one is so simple as to be ridiculous. For 2AAT knitting, wind the yarn for both socks before starting to knit. You start knitting the first side of the first one, and then then move on to the second sock, then you turn and do the second side for both. Don't wind-and-knit one on both sides, and then expect to be able to wind-and-knit the second! The link is for a written-up version, which I personally find easier to work from than a video, but I'm sure there are loads of YouTube vids showing this cast-on for those who prefer that medium.
2. figure 8 - this one's easy as well and gives identical results to the turkish cast-on to which it is very closely related. It's more fiddly than the turkish cast-on, because a) it's harder to tighten the loops before knitting the cast-on off of the second needle, and b) you need to knit the second needle's row TBL. In effect, you add the complication of doing a figure 8 to wind on, only to undo that complication when knitting. So, after trying this one once, I ditched it in favour of the turkish method. Again, for 2AAT knitting, make the figure 8's for both socks, then start the knitting.
3. Judy's Magic Cast On ("JMCO") - this one is much more complicated than the two above, and requires some serious studying of diagrams and/or videos. It's like learning the long-tail cast-on all over again. Check out YouTube for videos. In fact, when I use this cast-on, I'm pretty sure I'm doing it "wrong" because my second needle's stitches are all twisted in the wrong direction, so I need to reseat them prior to knitting (or, as is more usual, I simply ignore this and wind up with one row of inversely-twisted sts. Who cares? It's on my toes!). The advantage to this cast-on is that you don't have to worry about correcting the tension before you start knitting off the second needle. I use it quite frequently, although - as I've said - incorrectly.
These cast-on techniques are useful because they can be used as provisional cast-ons as well. In fact, there's an advantage to using them instead of more traditional provisional cast-ons, and that is that they overcome the problem of having one stitch too few or too many. That problem is explained in detail by TechKnitter, here. The JMCO is especially useful for this application, it works better as a provisional cast-on than the turkish/figure 8 because the cast-on stitches are more completely formed and it's easier to keep them on a string for pick-up, later.
But if you're looking for a quick 'n simple cast-on for socks or mittens, Turkish is the one to learn, IMO.