For the Euphorbia in question the best approach is to stand the cutting up on top of the soil. If you have to bury the end a little (maybe half an inch or an inch) to keep it stable, that's fine. With the strategic use of rocks or similar you can prop the cutting up from the sides. Another thing you can do is wait until it's watered and properly seated, then run a couple of chopsticks into the soil alongside the cutting, so as to help keep it vertical. Mainly keep the pot out of the way of traffic so it doesn't get bumped.
As for timing & exposure, this is what I do (and what has worked for E. trigona here). Cut carefully and do not get the sap on your hands, and especially not on your face, when you do this. Allow the sap to ooze out onto a paper towel and after a couple of hours you can handle the cutting without too much concern. The cold water trick speeds up this process. Leave the cutting in bright shade (lots of reflected light but no direct sun) for a week or so. Ideally use an area with good ventilation. I like a spot right outside our front door which is under an overhang but directly in the path of the prevailing winds.
Then set the cutting up for rooting mostly on top of soil in a smallish (not deep) container and water regularly (waiting for the soil to dry out each time). During the rooting process give the cutting bright light but little direct sun. Indoors ideally provide hours of daily sun during this period and beyond (indoor sun is not direct because a lot of the UV is filtered out by window glass).
These plants tend to have a seasonal growth pattern here and you may see the cutting sit there apparently doing nothing for months or maybe a year. Do not despair. And do not overwater. They tend to root first and grow later, so once you see new growth on top you have a pretty good indication the cutting is rooted and ready for a step up in pot size. From there on out try to provide strong light so the new growth is compact. These two plants are a few years old and they get about half a day of sun.
