Any pointers on this? I have all the skills in the "Mage class" leveled to apprentice (execpt Destruction, which is Adept. I'm at level 20). And yet, I cannot seem to achieve the same level of success as when I play Warrior or Thief. My mage has 200+ health and still gets insta-killed by Draugr Deathlords, even with Stoneflesh enabled. I find myself constantly running out of Magicka, as well as my spells (Fireball, Chain Lightning and Ice Storm) doing absolutely no damage to most foes. What am I doing wrong? I've tried on several occasions to play a good mage, but I can't seem to do it for the life of me.

Try a Battlemage setup, using heavy armor.

Take whatever set of robes/magicka ehanced equipment, disenchant it and apply the enchantment to a set of heavy armor, steel plate, deadric, etc.

You're spread too thin. You're probably at too high a level for your offensive skills. I think you need to concentrate on one skill at the beginning for survivability and then branch out later. I usually build Destruction to about 60 before working on other skills. You could substitute Conjuration or even Illusion if you prefer but you do need a main skill focus to get going.