Winterhold Radiant Quests Won’t Start

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.

unfortunately, the only way the level up is to increase those other skills. Sadly, you can't do the old "go to jail, lose skills, rebuild" from oblivion, because when you go to jail, it removes skill progress, not actual skill.

Fortunately, the game gives you easy ways to lvl up skills:

Complete the Mages Guild quest to get a spell that turns health into magic (equilibrium, i think). Weild equil in left hand (Alteration), heal in right (Restoration) and just cast until mana delpeted [Warning: This CAN kill you]. Make some potions to restore mana and health (Alchemy) before you do this and apply them when needed, and, when casting spells, make sure you're crouching in the middle of town (Sneak) [or in the courtyard of the College, next to the mana pool for faster regen]. Once you've maxed out restoration and alterartion, cast invisible (Illusion) and Fireball (destruction).

Next to combat, equip a Light armor Helmet and Torso, Heavy armor Gloves and Boots and your choice shield. [Enchant if you want] Learn the "Bound Bow" spell (Conjuration)(archery) and go kill enemies in the wild [Ensure to use bound bow during combat to get exp]. Once maxed, go and equip a shield [equip accesories to boost blocking abilities]. Find a companion, take them somewhere remote, attack them till hostile, and then block till maxed (block). [Easy difficulty means you won't take any bad damage]

Finally, Gold Making skills. Craft and enchant 100 iron daggers [look for the best profit enchantment], equip accesories to boost sell prices [or make them]. go to a weapon shop and sell them one at a time to the merchant (speech) [the game counts selling 1000 apple as one apple sold, so you'd get one lot of exp] [Each time the skill increases, exit out and enter again; it will adjust prices]. Once sold, make some clothing that boost pickpocket sucess and (if you have the perk) lockpicking. Then just go around and loot everyone at night, so you have to break into their houses (pickpocket)(lockpicking)

I didn't include one/two handed, because those just require using them.