So you wish to master the arcane arts? If you have the patience for it then you can become more powerful then any mere warrior or assassin. When you start the game you should focues your efforts on the skills of Destruction, Conjuration and Alchemy, you can use destruction to destroy your enemis(if you are having trouble then try combining it with a small blade or mace that will allow you to add some extra damage, but focus on your magic) you can use your conjuration magic to summon allies to help you in combat that will progressivley become stronger as you level up and gain access to new spells. Alchemy will be your supporting skill, as ingredients are abundunt and there is a alchemy table in every city you can quickly level up while creating lots of useful potions, you will also use it to create healing potions which will aid you in combat, and another use is to create potions that increase the effectiveness of your destruction or cinjuration spells or that reduce their cost for a while. Focus on these schools until they are at least around the 50-60 level, also make sure that you are constantly increasing you magicka and health stats each level, for a mage its good to go, magicka-magicka-health every three level ups. when you are around this level you can start branching out into the other schools of magic such as alteration, restoration, ect. At this stage just try to learn lots of new spells and increase your skill in all the schools of magic (not enchanting yet) while still raising Alchemy as Alchemy is one of the most useful skills in the game. Use your destruction and conjuration spells to dispatch enemies, at this stage you should be trying to minimise the amount of times you supplement your magic with a weapon. Focus on your magic. As you level up and increase your arsenal of spells you should be gaining access to the higher tier spells and your alchemy skill should be very high, now you can begin your path into Enchanting, find enchanted objects and disenchant them to learn their effects, keep doing this as it will not only increase the amount of spells you have available to use in enchanting but will increase you skill as well. when you have gained as many enchantments as you can (WHILE STILL LEARNING NEW SPELLS AND INCREASING YOUR SKILL WITH THE SCHOOLS OF MAGIC) you can use money made from sellling off potions you make to fund your enchanting, simply harvest a hell of a lot of souls and buy lots of jewellry and weapons and armour and clothing and enchant all of it.
(Gaining Azura's star is a particullay excellent way of ensureing you always have a soul gem ready to be filled when meeting a random bear or bandit, I suggest corrupting it into the black star so that you can harvest sentient souls as they are as large as grand souls and are in abundance within cities' if you wish you could murder citizens in one hold while enchanting in another, effectivley turning one hold into your "Soul Farm" while still being able to enchant in peace as holds track bounties seperatley.)
as you gain skill in the Enchanting tree you can increase the power of your enchantments, a good goal to aim for is to create a set of armour or robes, whatever you wish, that adds up to have a total of reduce magicka cost by 100%, e.g Robes that reduce destruction magic casting by 25% + Boots with same effect + hood with same effect+ gloves with same effect = 100% reduced destruction spell cast cost, etc. this will enable you to cast spells from your primary school of magic such as destruction or conjuration without using any magicka. This means that you can cast uber-powerful destruction spells for free while saving you magicka for other spells such as alteration or restoration spells, etc.
Keep levelling up your skills and learning new spells to increase your arsenal, if you have the Dawnguard DLC then I highly recommend becoming a vampire lord as it will open up a whole new level of power.
With enough effort and creative thinking you can become invincible as a mage! Hope this helped.