Separate answer because I feel it addresses a different point than the other answers
As stated in the other answers, you should of course teach basic clipping and leading techniques, making sure to emphasize the stuff that will get you killed (whenever teaching beginners I don't dance around this issue, "don't do this or you will die" ) basic anchoring and the like.
However, I like to stress, besides just teaching the technical details, is the proper APPROACH to outdoor climbing vs indoor climbing. Just because you can clip a dozen draws and lower off an anchor does not make you a good outdoor climber.
Things to stress
COMMUNICATION I religiously read ANAM and other accident reports on supertopo/climbing.com/mountainproject. There is a significant portion of them that are caused by improper communication. Let your belayer know EXACTLY what you are doing and what you are planing to do. Make sure your belayer lets YOU know what they are going to do if its outside basic belaying duties. There are multiple ANAM reports of people dying or getting seriously injured when the belayer assumes they are rapelling and the climber assumes they are being lowered off. Make sure to discuss communication strategies before committing to a climb, avoid the "oh fuck" moment when you've crossed over an exposed arrete, gone off route, and realize you are out of earshot and desperately need your belayer to take so you can back off
SYSTEMS before committing to a new belay system (whether going off belay and on your PAS at the anchor or switching to rappelling), check, double check, test, and double test the new system before going off the secure system to the new one. Always communicate any system changes with your partner(s). Always understand how the system works, especially if you are the one rigging. When taking newbies climbing I always make sure to describe what every component of the new system does before asking them to commit to my system.
EQUIPMENT Know how your equipment works inside and out before using it. Learn all the failure modes, features, how to properly rig it and its peculiarities. For example you would be surprised how many people do not know how to properly lower a climber on a top anchor system with a standard top-anchor ready friction belay device like an atc guide (Do you?). Do not trust that a device is intuitive, know it as well as the designer. Never use another persons equipment without examining it yourself (A well known youth climber recently died after all of his draws failed, the draws were a gift from his girlfriend and were improperly assembled). Never trust fixed gear without examining it, even on established and well used routes (There have been several high profile accidents where fixed draws on popular routes cut ropes during a leader fall). Know how to evaluate bolts and fixed anchors, always back-up fixed soft gear.
FALL DYNAMICS Know how a fall will load your anchors and your belayer, know the basic physics behind how the rope arrests your fall and what can cause that system to fail (z clipping is dangerous precisely because you shorten the dynamic length of the rope and significantly increase the fall factor). Half of all climbing fatalities are due to leader falls, and the majority of these do not have aggravating circumstances like gear failure or belayer error, falls are dangerous, knowing how to fall is important, knowing how to climb and place gear so your fall risk is minimized is even more important.
RETREAT Know how and when to bail off a climb, know how to deal with emergency situations, this includes basic knowledge of rescue systems (knots, tools, and techniques) and how to execute the. Too many people get stranded and call for SAR when they are more than capable, physically, of helping themselves had they practiced some basic rescue systems. This also includes planning a retreat strategy on longer multi pitch climbs and knowing how to descend once you get to the top. In many, many places the descent is significantly more sketchy and dangerous than the climb itself. Rappelling alone is probably the most dangerous thing you can do climbing besides leading.
There are far, far too many people that get profficient at clipping up indoor gym routes and then decide to try it outside thinking that cragging is just an outdoor gym. It's not, and make sure to stress that to any new person you take outside climbing. Just because you are strong enough to climb 5.12 doesn't mean you are a good climber outside, this applies triply so for trad climbing.
ONE FINAL WORD
Spontaneous gear explosion/failure is responsible for a tiny fraction of climbing incidents, of which the majority are related to fixed gear failures. The rest of the non-act-of-god (read:rockfall) accidents are due user error, your number one risk in climbing is you or your partner making a mistake (or several) so do everything to minimize the risk of mistakes, including your choice of partner. If you partner objects or is offended by you checking their systems, their gear, their rigging, or their belaying, DO NOT CLIMB WITH THEM.
