Chain slipping on newish bike
Learning some better shifting technique may solve this. It also could be that, despite your best efforts, the derailers are out of alignment.
Two questions:

Is the chain falling off on the front chainring or on the rear cogs? If so, is it happening on the inside or outside? (If it's happening in the rear and the chain is hitting the spokes, this can be an extremely serious problem that can be a danger to the rider. Other scenarios are not as serious.)
Can you tell us more about the bike? You mention a Shimano drivetrain; Shimano makes differing levels of components, and even top-level components won't work very well if a bike is assembled poorly. Is this a reputable brand of bike purchased at a bike shop, or is this a big box bike?

Shifting technique
This is particularly a problem with bikes that have twist-grip, bar-end, downtube, or other shifters that allow the user to shift through many gears all at once--or at least allow them to try; this will often pull the chain off the gears. The solution to this problem--aside from designing the bike to not do this in the first place, or having it adjusted more regularly--is to shift gears slowly, preferably one at a time. If you must shift quickly, at least slow down near the end of the shift!
You mentioned that the chain is coming off when shifting under load. On derailer bikes, you really are meant to shift while continuing to pedal but letting up a bit on the pedals. Shifting under load is not advised. I can't stress this enough.
A big part of proper climbing technique is getting into the habit of shifting in advance to avoid having to shift under load. (If you're stuck in a difficult gear, sometimes you can coast for just a second while you shift. You'll lose a little momentum, but you'll avoid walking the hill.)
Mechanical causes
If none of these solve the problem, you may need to face that your derailer adjustments may not have done the trick. I'd bring the bike into a shop and have a pro wrencher adjust the derailers.
Cable stretch tends to come about with use more than time, but it could be a contributing factor. A derailer service will include adjusting the cables.
There are other possible causes, although they're less likely than shifting technique problems or derailer maladjustment:
Chain stretch is actually the wearing away of the insides of the metal links, and it only happens with use. While a stretched chain can increase problems like this, I suspect that chain stretch isn't an issue. However, it's at least possible that the bike came with an older chain, and it's so easy to check for chain stretch that it can't hurt to do so.
Similarly, worn gear teeth can also cause this, but this is extremely unlikely to be the cause. (Checking for those is a little more involved than checking the chain.)
It's also possible that the bike has the wrong size of chain on it, or that the chain is too tight (or too loose).