This won't make all your troubles go away, but it will help:
1. Don't use the stock spells. Instead make your own at a spellmaker (like the Altmer lady in the Balmora MG). Make minimum and maximum magnitude equal if you want a reliable spell. Make sure duration is at least 2 seconds. This is because the first second costs double. So a spell that does 20 Fire Damage for 1 second and a spell that does 10 Fire Damage for 2 seconds are both equally effective in combat, but the second one costs 25% less Magicka to cast. And 7 damage for 3 seconds would be 33% cheaper to cast than 20 damage for 1 second.
2. Learn the resistances and weaknesses of your various enemies and use appropriate spells. Frost Damage and Poison Damage don't work very well against undead and casting Fireballs at Dunmer isn't a very good idea either. You can use the UESP http://www.uesp.net/wiki/Morrowind:Creatures to aid you. Note also that Resist Magic doesn't protect against all spells, it only protects against "pure" magic effects like Damage Strength, Absorb Health, and Charm. Basically it's used to protect against anything that doesn't have it's own resistance. So Fire Damage, Frost Damage, Shock Damage, Poison Damage, and Paralyze will all ignore it.
3. Fire Damage and Frost Damage are the cheapest damaging effects to cast. Shock Damage is more expensive, but resistance against it is less common. Poison Damage is overpriced, don't use it. Absorb Health and Damage Health have the same Magicka cost and are both countered by Resist Magic so Absorb Health is obviously much better. Absorb Health also has the benefit that it's harmless if it gets reflected back at you (as long as the magnitude modified by your Resistance or Weakness to Magic is lower than your health). I'd mostly stick to Fire Damage, Frost Damage, and Absorb Health. The first two are cost efficient (assuming the target isn't resistant) and the last one is safe (and relatively few enemies resist it).
4. An Absorb Health spell with low magnitude, long duration, and large radius is an excellent spell to cast if you get swarmed by multiple opponents. If your spell absorbs 2 points of Health and you hit 3 opponents you will get healed for 6 points per second.
5. Save your spells for "worthy" foes. It's a waste to spend Magicka on fireballing weak enemies. Spending magicka on a bound weapon which you then use to kill these weak enemies however, is fine.
6. Get into enchanting. Make rings that shoot fire and frost and absorb health. Unlike your Magicka the charge of enchantments regenerates over time and since rings only have 0.1 weight even a mage can easily carry dozens of them without weighing him down.
7. A 1 point Levitate on self for 5-8 seconds is a great way to reach some places that would otherwise be difficult to get to. A 1 point Levitate spell with a long duration on touch or target can be used to slow down enemies. And make a custom Levitate spell on target for 1 second. It only costs 1 Magicka to cast and if you shoot Cliffracers with it they will loose their ability to fly and fall to the ground.
8. If you ever have to get into melee use a Bound Longsword. It's not expensive to cast, has great damage, and fortifies your Long Blade skill by 10 points.
9. Shield, Feather, and Burden all svck because their casting cost is too high. Use Bound armor, Fortify Strength, and Drain Strength instead.
Edit: When you are making your own spells at the spellmaker pay attention to the Magicka cost and casting chance that are displayed in the upper right corner of the spellmaking window. Overlooking those two numbers is a fairly common mistake which can lead to an otherwise great spell that you simply can't cast reliably. You'll probably want to stick to spells that have at least 80% casting chance.