I'm fairly new to Skyrim, after running around and completing a few quests I was still only able to cast novice spells. I though I'd go back and complete the college quests and see if that got me a higher level of spell. I have been made the Arch Mage got his room key and everything. How come I still only have novice level spells?
Do I have to buy the training of the other college members?
I generally use destruction and restoration magic alongside melee combat.
I just started to use magical weapons so use the soul capture spell.

Any tips on increasing my magic level?

Thanks

Using skills gains you XP, and getting enough XP gets you a skill gain, and when you get enough skill gains you level up, and then choose health, magicka, etc. You also get a perk. If you use the skill enough, and you have the right skill level, you can choose a perk, which, in the case of Destruction, lets you get a perk which reduces the magic cost of your spells. You can get higher level spells by buying spell tomes from court wizards or the College teachers. Youy can learn the respective spell from the tome.

Yeah, the College of Winterhold quetline actually does very little to help with your magic, strangely. Even if you're Arch Mage. If you want to improve your magic, some of the standing stones like the Atronach Stone and Mage Stone can help a lot. Though I would suggest the Mage Stone, as it makes your magic skills increase faster.

Get the mage stone it will increase your magic skills faster. And you can train with the teachers to increase your magic skills, you have to pay for it though.

The Apprentice Stone increases your magicka regen, which can be helpful, but it makes you extremely weak to magic. I wouldn't recommend it to beginners,

I just got Dawnguard and my Dunmer Necromancer/Lich/Vampire Lord (she'd been around a long time when I played her in Morrowind) is going to use the Apprentice Stone. The weakness adds that extra bit of challenge, and, for me at least, it adds so much intensity (I play DiD with 95% of all my characters, so far this one has survived Morrowind and Oblivion, now will be another test).

I really think that there should be certain skill requirements for archmage rank (it would have been a simple matter of keeping Mirabelle around), but I didn't write the game.

Not much changes once you reach archmage rank. However, you do get a cool set of quarters in Winterhold, complete with herb garden, enchanter, alchemy lab, and tons of secure storage. And it's all free. If you're using the Lover stone (which is my preference) and getting a constant bonus, you can't get a well-rested bonus, so who needs to buy a house? You've got digs that are ten times the size of the local jarl's.

Just buy spell tomes that aren't novice level, and you will have non-novice level spells.