Mages Guild quests are generally pretty straightforward affairs with a couple of notable exceptions. A few require you to be able to cast specific spells. You must use the standard (circinate) form of the spell rather than one created in the spell-maker.
Retrieving ingredients (a common quest for the Mages) can be tricky because the ingredient (especially blood) looks like regular dungeon dressing. You have to be very sharp-eyed to find it.
"One of our books is missing..." (any rep): ask around town to get the name and location of an informant (usually at an inn) who will give you the name and location of the thief, who will have the book on his body.
"I do have a small task that needs to be done..." or "I do have a small job that nobody else is willing to do..." (any rep): find and retrieve the ingredient from the dungeon. You should get a dungeon marked on your map as a reward.
"...if you would be willing to chase down an escaped experiment..." (any rep): hunt down and kill the imp (steel weapons or better to hit it).
"...I have something that I need done that I don't really want anyone else in the guild to find out about..." (any rep): find the book/scroll in the dungeon. Don't read it ("use" it) unless you want a surprise.
"We need you to get a rare ingredient that is hidden away in a nearby dungeon..." or "One of our more brilliant students is absolutely mad for something called <ingredient>..." (any rep): find the ingredient in the dungeon.
"...is there anything more exasperating than the discovery that the mummy wrappings you were planning on using in a potion have gone all sort-of green?" or "...I'm working on a formula that calls for a "goodly piece of <name of mummy>..." (any rep): find and kill the mummy in the dungeon. It will have mummy wrappings on a green background as part of its treasure.
"Noble <name> has sponsored the Guild in many of our enterprises..." or "If you are capable of casting the standard circinate spell Sleep..." (any rep): the only tough parts of this quest are (1) having enough magica to cast the Sleep spell and (2) having enough magica to cast the Sleep spell AND the Open spell you will need to get through the locked doors. Do not use anything other than the standard "Sleep" spell purchased from the guild or you will have to fight the client.
"As the premier research institution in the Empire of Tamriel..." or "The Mages Guild of <town> has an exciting opportunity to benefit from the work of..." (any rep): the Dark Brotherhood also wants the research. They will usually send you a letter telling you to bring it to them, but this part is very buggy and the letter doesn't always arrive before you have delivered the papers (so your rep with the DB takes a nosedive). If you take it to them, they will sometimes (depends on the random numbers) give you a dummy document to deliver instead, but not always.
"...I need someone who can cast the Open spell..." (any rep): the box is in town. Casting "Open" while it is in your inventory can generate one of several events:
"...I am perplexed. Perhaps you can help me with some research..." or "...if the internal azimuthal vectors really do cross at the anterior pole..." (any rep): this is a very confusing quest. You are given two places to look. One has an NPC and the other has a fragment of a catalogue. Sometimes the catalogue will help, sometimes not. Sometimes the NPC can help, sometimes not. The NPC will always want you to fetch some ingredients from a town. The most consistently successful strategy is to head for the NPC and run the errand. If the NPC does not know where the item is, she will offer to teleport you to the room where the catalogue fragment is. If she does know where the item is, she will tell you (and give you something for your trouble in running the errand). The most difficult part of this quest is doing it before time runs out.
"...one of our experiments has prematurely evacuated our laboratory..." or "Actually, we have had a bit of excitement in this dull old place lately..." (rep 10 or more): go to the dungeon and kill the atronach.
"The Mages Guild is in need of someone to guard the <magic item> tonight..." or "The Guildhall -- how should one put this..." (rep 10 or more): stand guard in the guildhall from 12 to 3. Four Thieves will show up. Your reward will usually be a cheesy gemstone. If you have a good cushion of rep with the guild, save the game right after accepting the quest and take care of the Thieves. Then go pick up the item (blows the quest) and identify it. If it's a good item, take your 2-point rep loss and keep the item. If it's not so hot, restore the game and play out the quest the way you're supposed to. The Thieves will show up sometime between 12 and 1.
"You are loyal member of the Mages Guild, are you not..." (rep 11 or more): this is the start of a series quests that will automatically activate if you do not kill the rebel mage. If you kill him, you will get a magic item for your reward. If you spare him, he will send you a message a few weeks later.
"I am <name>, servant to the mighty <mage>...." or "It is a very great honor, my dear <PC Name>..." (rep 20 or more): your guard duty starts as soon as you accept the quest. Three Nightblades show up after an hour and you get a magic item for your troubles. Easy quest since the Nightblades don't use magic on you.
"Doubtless, you have heard of the crazed wizard <name>..." or "The people of <province> have been demanding that the Mages Guild take a stand against the mad wizard.." (rep 20 or more): standard dungeon crawl with a magic item as your reward. It also nets you a 10-point gain with the ruler in the province where you accept the quest. Very handy if you've been beating up on Gothryd's guards or if Eadwyre is ticked at you (like he's ever not ticked at you).
"Are you <PC Name>? The <Guild Rank>?" (rep 21 or more): there are a couple of ways to play this one out. The first thing you must do is go talk to the Acolyte at the temple and then talk to the noble. From there you can either fetch the Oracle from the temple the Acolyte told you about or you can fetch the noble's lover from the dungeon the noble told you about (or you can do both if you're feeling extra frisky). You get a boost to your rep with Akorithi in Sentinel for delivering the Oracle and some gold for delivering the lover. Like I said, if you're feeling extra frisky, do both to collect both rewards (rescue the lover first, then go retrieve the Oracle).
"We need someone to kill a rogue imp..." (rep 30 or more): travel to the town, bop the imp, (steal the paintings from the crates in the back of the Palace), report back for your gold.
"I'm looking for a mage powerful enough to cast the spell Banish Daedra..." or "Banish Daedra is one of the most difficult..." or "...It is a great irony that summoning a daedra is a relatively simple matter..."(rep 40 or more): a tough quest at high levels when you run into Daedra more frequently. Up until that point there is usually only one Daedra in the dungeon. You must damage the Daedra, read the scroll and then cast the spell. If you use the scroll on the wrong Daedra you will get another Daedra appear in the area, so you will have to fight two. The best strategy I have found is to save the game as soon as you see a Daedra and then follow the banishing procedure. If it works, you're done. If it doesn't work, reload and keep going. Be aware that critters who are in melee range at the time you save a game often lose their treasure when you load that save.