Continental knitters shouldn't be afraid to try to knit with two colors in one hand because it is done very similarly to how they knit with one color.
With this method you tension both yarns over the left hand index finger. Here's how I do it.
By stranding the yarns as shown (one going over and one going under the middle finger) I can try to avoid the two yarns ending up too close to each other on my index finger. (A strickfingerhut can also help keep the yarns separate but I'll talk about using one tomorrow.)
The yarn lowest on the finger is knit normally - by taking the right hand needle, putting it through the stitch on the left hand needle knitwise, then going over or under the lowest yarn with the right hand needle and pulling it through the stitch. I don't think it matters much whether you go over or under the yarn to grab it. Here I am going under the green yarn.
To knit the color held higher on the finger, nearer the fingertip, you take the right hand needle, put it through the stitch on the left hand knitwise, reach beyond the black yarn and grab the green yarn and pull it through the needle. I've tried grabbing this second yarn by either going under or over the yarn and I've found that using the needle to grab the color by going OVER the yarn is faster for me.
Here the black yarn is pulled through and I'm back to where I started.
As I said, the biggest difficulty I have using this method is how to keep the two yarns separate enough on my index finger so I can easily choose one color to knit. I do notice I lift my left index finger often to help tension the two yarns. Or I can use a strickfingerhut/knitting thimble. More on that tomorrow.