Yes, the trick is to move the yarn BETWEEN the two needles when moving it to the front for a purl stitch, or to the back for a knit stitch. It seems you might be accidentally taking the yarn over the TOP of the right needle instead of between the two. Those extra yarnovers could be helping to pull the stitches out of whack too... eliminate the yarnovers and the stitches might be less prone to twisting round the needle.
The way its twisted around might also be because when you have just cast on, the first row is a little more fiddly than later rows. The stitches can tend to rotate round the needle. Just examine the look of the row of cast-on stitches before you start to knit. Then after each stich is knit, take another look at the stitches remaining on the left hand needle and make sure that they look how they did when you'd just cast them on, nice and straight and even, and haven't turned round the needle. Once you get past that first couple of rows it becomes easier.
What casting on method are you using? Long tail? If so... I know a lot of people might disagree with me... I find it more fiddly to do the first row after that sort of cast-on than it is to continue on after using the cable cast-on.
If you haven't tried the cable method, give it a try. I'm a big fan of that method, and it might be a good one to try even if only in your early stages of knitting.