Sure, all DM's worth the name train their players. If you've ever killed a PC, then you are training your players, because nothing teaches a lesson like rolling a new character.
Lessons players learn in my games tend to be:-
1. You can't beat every encounter with swords and fireballs. Some monsters are too tough for you to fight, but that doesn't stop me using them.
2. If you're losing a fight, you CAN surrender. If you throw down your sword and shout, "My ransom is twenty thousand gold pieces!" most monsters will suddenly become amenable to taking you prisoner and letting you keep your eyeballs.
3. You need to declare when you're searching for traps. After I've told you to roll a saving throw, it's too late to say "Oh, but I search for traps first."
4. Do not attempt to charge at a lich in its lair and hack its head off, it will have countermeasures prepared for this. More intelligent statements of intent are a survival trait.
5. Forget you ever heard the phrase "BBEG." If you attempt to understand what's going on in my game world in terms of facile concepts like "BBEG" then you will fail totally to grasp the big picture.
6. Act in such a way that your other party members can trust you. My environment design assumes that the party will act in a coherent, co-operative and well-thought-out manner and I will not reduce the difficulty of the next encounter simply because both the clerics have gone off in a huff. Chaotic Evil characters can live, but Chaotic Stupid characters are doomed, and they might well bring the whole party down with them.
7. Before pointing the wand at the bugbears and shouting the command word, find out which end of the wand the fireballs come out of. (This lesson was repeated quite recently *grins*)
8. Do your homework. Visiting the library does have a chance of finding records of a previous expedition into the vampire's crypt, and you can thereby discover that yes, he does have several coffins.
More intelligent DMs would be astonished to learn how often so-called "experienced" players fail to appreciate these basic D&D lessons when they come to my table...