Many modern stickshifts have hill assist, so if you're fortunate enough to be teaching on a model with this, you either want to temporarily disable it (if possible) or at least communicate the right way to do it and what might happen if they're going to driver an older car.
I don't know if it's just the couple of cars I've driven do it badly or if they're typical of hill assist on all cars, but I really hate this feature. It always seems to keep the break on longer than I want, and on a couple of occasions even causes me to stall because I tried to accelerate away and couldn't move. Now I usually put the hand-break on because I find it easier to find the biting point [0] and move from there (pretty much in one smooth motion, not needing to pause at the biting point).
I've never bothered to think more about it, because it doesn't cause me any trouble doing that, and afaik there's no harm to the car doing it this way, but maybe somebody has advice on how to use hill assist without finding it so annoying? (Or news that in most cars, hill assist is better than the ones I've used?)
[0] Biting point being where you pull your foot gradually off the clutch, not all the way but to the point that the front of the car bonnet starts to rise slightly, which is the point that you can remove the hand-break as you keep declutching without rolling backwards down the hill. Was one of the trickier things for me when learning to drive, but incredibly easy once you're used to it.
I too just use the handbrake since it feels natural after growing up with it. By the time I'm accelerating I mostly don't notice if any hill-assist was engaged - but this may be a factor of my particular car, and I'm unfortunately unaware of how it feels in some other modern stickshifts.
I haven't driven a manual with hill assist, but indeed this is a difficult thing to master especially on an unfamiliar car. It takes a while to get used to the pedals on any car, much more so the clutch. Drifting backwards down a hill is a scary thing to happen at first!
I find that automatic drivers aren't used to engaging with the handbrake, which is the real secret to starting up a hill.