So is Resist magic better than resist frost or fire? Meaning, does resist magic give you resistance against fire, ice and other magic damages?

Yes. I'm not sure if Dragon Breath counts though.

I put Resist Fire and Resist Shock on boots and then carry a Ring with Resist Magic and Resist Frost and a Necklace with Resist Fire and Resist Shock (Nord characters).

Edit:
You can also do

I like magic resistance a lot. It gives protection from magical damages, such as fire and frost breath from dragons and magic shock attacks from enemy mages. Best thing about magic resistance is that you can get a lot of it without wearing any enchanted gear. Bretons get 25% magic resistance. You can get 15% from becoming an Agent of Mara. You can get up to 30% from from the 3 magic resistance perks in the alteration tree. You can get 25% from choosing Lord Stone. They are capped at 85%. With magic resistance alone you become very sturdy against any non-physical attacks. You become practically invincible from magical attacks if you stack Elemental resistance on top of it.

If a frost attack does 100 damage. 85% magic resistance cuts it down to 15 damage. and if you have 85% Frost Resistance on top of it, it becomes 2.5 damage. Completely harmless. 97.5% damage reduction.

I used to use build that gave me 85% magic resistance and 85% fire, frost and shock resistance. It was so good that the game became too boring. I now use 65% magic resistance (from 30% alteration ,15% agent of mara, 20% from enchanted shield) 50% frost resistance from Nord blood, 37% shock and fire resistance from dual enchanted boots. So 100 damage frost attack will do 17.5 damage. I still carry around a ring and necklace that will boost magic resistance to 85%, frost resistance to 85%, and fire and shock resistance to 74% if I feel the need. Just in case.

You typically are able to get larger quantities of resist elements than resist magic.

Case in point: Bretons may have 20% resistance to all magic, but Dark Elves and Nords have 50% resistance to Fire and Frost, respectively.

Yeah, I think it comes down to enchantment cost. You can probably place a higher resistance percentage on an item if you use the element resist rather than a blanket magica resist. The downside to this is that you would use up a lot of enchant slots on your gear in order to cover all your bases. Also, there is no "resist absorb health/stamina/magica" enchantment so you would still be vulnerable there.

Magic resistance is better since it covers you from all 3 kinds.

That means you would need 75% fire resistance to (on average) protect you as much as 25% magic resist.

So if I can enchant resist frost or resist magic, I would be further ahead to enchant the resist magic since it covers all elements right?

Yep. If you get your magic resistance high enough, you'll only need elemental resistances on top of that against those gatling mages.

With maxed magic resistance and just 37 to fire and frost on my boots I can take care of a dragon priest no problem at all.

Uhhm... that's not how math works.

25% Magic resist stops 25% fire damage.... and frost damage, and shock damage.

75% Fire resist stops 75% fire damage... but not frost or shock.