To become a Mages Guild member, your character doesn't have to be a strict magic user of any sort. The missions' structures usually suggest using magic, but in very few places you are forced to do so without any other way around (and even there you can always get away with it using scrolls). And yes, that means that you can actually become an arch-mage while not being virtually a 'mage' at all.
Joining this Guild is not a problem, just as with the Fighters Guild - you are only expected to have a clear criminal record. Talk with one of the branch leaders in one of the Mages Guild outposts around Cyrodiil. To become a full member and gain access to the Arcane University, though, you have to get a recommendation from each of the 7 branches. In other words, there are 7 initial quests to be performed.
Quest from: Carahil;
To get a recommendation from Anvil, you must solve the mystery of merchant assassinations north of the city. Go to Brina Cross Inn and talk to Arielle Jurard. She'll tell you to hire the room and wait for her there. Talk to innkeeper Christophe and tell him you're a merchant. When going upstairs, you'll be approached by Caminalda, who'll warn you about the bandits in the area. Go to sleep, and Arielle will wake you up. She'll fill you in her plan. Go to sleep once more, and once you wake up go along the road towards Kvatch.
After a short walk, you'll be ambushed by Caminalda, who turns out to be the murderer. Arielle and the other battlemage will help you deal with her. Once you're done, go to Carahil for your recommendation.
Quest from: Kud-Ei;
Kud-Ei from Guild's branch in Bravil will tell you to talk to Varon Vamori and retrieve a stolen staff from him. He's somewhere nearby, as he's with the Guild as well. Use some Speechcraft or Charm on him and he'll tell you that he sold the staff to Soris Arenim. He lives in Imperial City Talos Plaza. Talk to Kud-Ei and go over there.
When you raise the Dunmer's disposition high enough, he'll offer you to buy the staff for 200 Gold. Do so, and Kud-Ei will gladly give you the recommendation. She'll also teach you the spell Captivate.
Quest from: Jeanne Frasolic
Jeanne wants you to find a missing mage named J'Skar, who is probably trapped in other dimension of sorts. Seek help from Volanaro, who'll agree to solve your problem in exchange for a favor. You just have to sneak into Jeanne's room and steal a Manual of Spellcraft from her desk.
That's an easy task. Once you're done, Volanaro will tell you to meet him in living quarters downstairs at 10 P.M. Wait until this hour and Volanaro will just dispel the invisibility spell on J'Skar. Those ruffians were just making a fool out of their boss, who apparently doesn't know a thing about magic. Anyways, you can now tell Jeanne that your mission is complete and receive your recommendation.
Quest from: Falcar;
Falcar orders you to find a particular ring, which has been thrown into the well behind the Guild building. The Argonian named Deetsan hold the key to the well. Wait until Falcar leaves the guild and she'll tell you that this mission might be dangerous, and will teach you the spell Buoyancy. Leave the building and enter the well.
Deetsan's spell turns out to come in very handy, as the well is (what a surprise) filled with water. The ring can be found by dead Vidkun's body.
It turns out that while you were down there, there was a fight between Falcar and Deetsan. As a result, the former denied your recommendation. Follow the Argonian's suggestion and enter the cellar to search Falcar's room. You'll see two Dark Soul Gems in there. When you show them to Deetsan, she'll tell you that she'll personally take care of your recommendation. You'll meet Falcar again - far, far later on.
Quest from: Teekeeus;
Teekeeus, Chorrol branch's leader, wants you to find a book "Fingers of the Mountain". An elf called Earana, who usually hangs round in The Gray Mare, would also want a peak on that ancient artifact. There's a long journey ahead of you, towards the place marked on your map. Watch it, as it gets really steeply in there.
You'll reach an Ayleid shrine. The book is by the burned body laying around. If you get it back to Teekeeus, he'll immediately give you a recommendation. However, if you decide to give it to Earana, there's more to get. Wait 24 hours to give her time to work on the book.
She will give you the "manual" of the shrine you've just seen. To get some of its power, you need a Welkynd Stone - these are always found in Ayleid ruins (consult out World Map section). When you have it, go to the Shrine and cast the spell Shock on it. You'll learn a new spell: Finger of the Mountain, which is a stronger version of standard Shock.
OK, but what about our recommendation? Teekeeus won't be too happy about you giving the book to Earana, but will give you a second chance. You have to steal it back from her. Simple as it gets - just sneak into her room in The Gray Mare overnight.