My favorite times to be a pure mage is very early on and very high leveled. Early on your very weak and everyone has swords and axes. Occasional archer.
Later on at high levels you've probably thought about your glass weakness and thought of ways to overcome them.
Alteration has some great passive abilities for countering fellow mages.
Illusion is great for a mage trying to be stealthy or non violently ending a fight. Possibly getting people to fight each other instead of you.
Destruction is simple straight forward. Means of killing.
Conjuration is amazing. Hardly any confrontation when you have two strong meat shields fighting for you.
Enchanting is great for enchanting a set of armor for your mage or robes. My mage had clothes enchanted to increase magic resistance and decrease magicka cost for destruction and conjuration. I can basically spam expert destruction and conjuration all day long. This is also largely due to my large magicka pool.
I like alchemy for merchant reasons. I never use alchemy for myself except for cure disease and health potions when my life and magicka are both low.
I also gave my mage an additional 100 points in hp via level up so he cant be straight up one shotted.
Restoration is the best. Not only can you slay any undead with dawnguard dlc added restoration spells, but you also obviously keep yourself from dying. Grand healing is stupid amazing. Not only does it heal yourself but it heals your nearby allies at the same time. Restoration also has really cool passive perks as well.
That's my reason for liking pure mages of skyrim. My biggest threat would have to be archers, but later game I can use my wits to counter them.
Personally my least favorite classes would have to be two handed warriors. I've never played two handed warrior. Always have something different in both hands. Another axe, a spell to go with my sword. I ran a cannibal thief werewolf archer once. It was okay. Not as fun as my wizard, vampire, or orc but okay.
I mostly enjoyed my vampire lord because of all the evil things he could do. The transformation was mostly an added benefit. Most of the time my vampire ran around bashing faces in with the mace of mulag bal and ripping souls from them with the black soul star. Truly a dark sadistic individual. He wore ebony mail and had ebony armor to match minus the helmet. I gave him a the aetherial crown as well as the jagged crown(pickpocket method).
The orc was gun because I hardly ever did missions. Most of the time I travelled across skyrim mining caves, selling wares, and building all three hearthfire homes.
The dragonborn dlc added so much cool stuff for my mage and for my evil characters. I loved going to solstheim in morrowind as well as skyrim. Herma Mora's books gave beautiful secrets.
I never play as khajiit or argonian. Not much style. I tried a redguard once but found myself repeating the same actions I did with another playthrough so I cut him short.
Don't care for bretens or nords very much.
Bosmer and orc are okay, I don't care for dunmer but they are okay to a lesser extent.
The imperials are my favorite human race. Pretty versatile. Fun battle mages. I'm also partial to them because I love the imperial city in oblivion. I spent countless hours in the imperial city of cyrodiil. I also kinda like the silver tongued entrepreneur.
In my experience playing skyrim, being evil is more fun than being good. So many good quests get ignored if your mr goody two shoes.
My mage was lawful neutral so his fun came purely from exploring and his lust for power.
I always play on easy mode, but I have played harder modes just to see how well I do. I always revert back to easy because being a godlike powerhouse is more fun to me. I'm not looking for a challenge. If I wanted a challenge I'd go outside and practice goju Ryu karate. Though funny enough I can appreciate in game challenges so I do occasionally crank it up a notch.