Well, I know that hagraven is part of a quest. And yes, it was very hard for me as well. Without my flame atronach serving as a distraction, I would have never beat it at that early level.

And I know from experience playing a mostly mage character is harder. I have two playthroughs right now: My main is an Argonian warrior dual wielding swords that is also very good at conjuration.

I also have a high elf mage who uses all types of magic (mostly destruction and conjuration though), but my character also uses a shield. Magic plus shield is a cool combo in my opinion. For the first bit of the game being a mage was easy. After that it starts being quite the challenge.

What type of armor are you using? And is it heavy or light?

~U_K~

I've heard that destruction magic is really underpowered, especially at higher levels; that's pretty disheartening. I might have to give magic-sheild combo a try.

I'm wearing some from of robe that decreases magic casting cost or something, as well as heavy boots, pauldrons, and helmet.

Here's a tip: Get good at enchanting. For me, whenever I find something with an enchantment on it, I disenchant it and get a level up. If you really want to level it up fast, find the shrine of Azura and finish that quest. For finishing that quest you have a choice of either Azura's star or the black star. Azura's star is the "good" choice I guess, but it doesn't matter as far as I know. It's not as good as the black star because the black star can get human souls as well as animal souls whereas Azura's star can only get animal souls. ANYWAY, these are grand soul gems that never break, and to get your enchantment up quickly, I would suggest enchanting a sword with soul trap and then you'll have infinite soul gems for it and each time you recharge the sword, your level will go up some more.

Once you have a high level in enchating, you can get the same bonus that you have for your mage robes and put it on any armor. Technically you can do that right now, but because of your lower level, the enchantment will not be as powerful.

~U_K~

Is there any way to fill soul gems besides using a soul trap weapon?

I had one before, but switched to a frost sword because it's more useful in battle.

There is a spell called soul trap which affects the target for 60 seconds.

~U_K~

Give your companion all of your soul gems and a good soul trap weapon. It works wonders.