Some great points have already been made but I would also like to add a few. Magicka is one of the biggest limiting factors for a mage and as such Alchemy and/or Enchanting are needed if you want to play a pure magic caster. A warrior can still swing a sword if they have no stamina, a mage can run around a room for ages to wait until they can play the game again and that's not really fun.
Enchanting lets you make apparel that either reduces magicka cost of a school of magic (up to combined total of -100%) and/or increasing max magicka and magicka regen. Remember that the more magicka you have the faster you regen (magicka regen is 3% of your max magicka per second, 0.99% per second while in combat) so finding a balance of regen/max magicka/reduction of cost will be something that you will have to play around with yourself. The soul cairn is a great source of souls and soul gems for levelling Enchanting.
Alchemy lets you make really strong magicka potions for super cheap, and if you have Hearthfire (or better yet the Goldenhills plantation CC) you can shit out unlimited potions for free and little effort (also makes levelling way easier. Alchemy xp is based on the value of the potion made. Creep Cluster + More Tapinella + Scaley Pholiota if you are going to grow ingredients for the xp grind)
As mentioned by
@Friendly Primarina magic scales really poorly in the game. The problem is that magic seems really powerful in the beginning (as you mentioned) but it falls off hard so it kinda "traps" a lot of players. You can make a good mage build in vanilla but it's going to take more work then a warrior or stealth build. If you aren't against modding I would highly recommend finding a simple magic scaling mod. I personally use
Ordinator which is a massive skill overhaul that adds magic scaling and a ton of new perks to each skill tree, forcing you to specialise in skills leading to better class RP imo. Ordinator does offer some nice synergy between mage skills. If you do use Ordinator keep in mind you can make very powerful characters in the late game so it might be worth playing on a harder difficulty than you are normally comfortable.