Does the trainer you use make a difference

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.

You can only train 5 skill levels per (character) level.

So if you are level 10 and train destruction at a trainer you can train 5 times. Then you have to get to level 11 to be able to buy skill levels at a trainer.

That is the only restriction I know of (being lvl 62 atm).