Does magic damage add on to a weapon’s damage or is it a part of it

Making it "easier to cast" is relatively straightforward - it reduces the Magicka cost of the spell, meaning you can cast them more often.

This is similar to how things worked in Oblivion. (I'd link you to the UESP page documenting this, but the Site's still getting hit pretty hard from all the Skyrim traffic)

Each spell has a "base cost", which is modified by the following formula:

spell cost = base cost * skill multiplier * perk multiplier * equipment multiplier * dual cast multiplier
skill multiplier = 1 - (skill/400) ^ 0.65

This means that at 100 skill, spells cost 60% of the base cast (further modified by perks and equipment).

This isn't combat related, but as Affe has pointed out below, increasing your magic skills is required to buy high-end spells from vendors.

Thankfully, Skyrim does not suffer like Oblivion from all those unintuitive effects.

The one exception is magic ABSORB (not resist). It is available from the Breton's activated racial ability, the Atronach standing stone, and an Alteration perk. Even then, it only causes your conjurations to fail, and does not affect your healing magic, potions, etc. My Breton character simply doesn't conjure anything and enjoys full effect from everything while laughing off mage NPCs. If you want to conjure things and resist magic, simply avoid magic absorb and steer yourself towards hitting the 80% RESIST magic cap.