In all of my previous Morrowind playthroughs, I've usually played warriors. My strategies would involve gearing up in the most protective armor I could find, bashing away at enemies with swords, and then using various means to heal the damage so that I could do it again. As such, I never really worried about the magical capabilities of my enemies. Sure, the ones using Paralysis or Damage Attributes were annoying, but as soon as my Strength and weapon skills got high enough to reliably stun-lock enemies, it was only a matter of closing the distance. And that was only in situations where I couldn't dodge the spells until they run out of magicka.

Now, however, I am playing a magic-based class, and part of the reason I'm enjoying it so much is that I have to be constantly aware of the magical abilities and effects of my enemies, and take careful steps to counter them. I've been largely successful. But one effect I just can't seem to counter reliably is Reflect. Most daedra have it. A couple of named NPCs have it. Those riekling raiders definitely have it. For a very long time, I figured that a mage wanting to cast spells on something would first have to use Dispel in order to break the effect. Having been trying that, however, I very often have my 100% Dispel spell reflected back at me. Even when it seems to work, the Reflections still happen.

So my question is, what are the classic mage strategies for dealing with Reflect? Am I missing something incredibly obvious? Or is it just always going to be a pain in the rear when you're trying to cast spells?

Reflect is always a problem. One thing is to use absorb health spells on reflecting creatures so that at least it doesn't hurt you if reflected. This is considered an exploit by some and MCP has an optional "fix" to stop it. Personally, I usually have trained marksman enough to use a bound bow when facing such creatures.

yeah reflect is what turns me off from playing casters in morrowind, i usually have a back up weapon skill like short blade or blunt to counter dremora and reiklings. i suppose using a summon would be the best way to deal with them if you don't want to use weapons.

Absorb health (as stated above) is useful as it won't kill you if it gets reflected, i don't view it as a bug so i don't change it with mcp myself

As you have surmised, you cannot play a mage like a warrior who uses fireballs instead of weapons, unless you want to have your face burned off. Playing mages calls for some strategy, particularly when dealing with refective opponents. There are a few good suggestions above. I will see what I can add.

One option is to build up resistance for the type of spell you are using. So, if a Dunmer mage, cast fire spells against reflecting opponents, so if they are reflected back at you it will be reduced by 75%. I am pretty sure that the resistances stack, so your dunmer mage could cast 25% resist fire (or obtain it through an enchantment or potion), and then overwhelm a reflecting target with fire spells without fear that it will harm you.

Altmer have weaknesses to elemental magic, so they are the hardest to play against a reflecting target, since it takes a lot of resist to overcome those weaknesses. In that case, using the Absorb Health works (provided you don't "fix" it with the MCP). Other strategies are to use summon spells and let them do the work, or if you are facing a group, to use a command spell on one of them. A command spell won't hurt you if reflected.

Another idea is to cast a powerful weakness to element on your opponent and if that is reflected, wait for it to wear off before casting your next spell. Once it is worn off, try again until the weakness sticks. When that happens, launch a elemental damage spell that is calculated to be weak enough not kill you if reflected. The power of the spell will be multiplied by the weakness if it bypasses the reflection and may be enough to kill. If not, cast another one before the weakness wears off. If it reflects, it won't kill you and you can take a healing potion before trying again. this strategy works best when it is an element that the target is already weak to (or at least not resistant to) and when it is not an element that the target is casting at you (or the reflected weakness could be really bad).

These are just a few examples. There are numerous strategies for dealing with reflecting opponents, since the game's magic system is so varied.