First things first: Nobody sees the world like a mage does. Starting at level 17, your mage can learn the Teleport spells for all of your faction's capital cities. Starting at level 42, you can learn the portal spells and then cast the wrong one and leave your friends staring at the spot where you used to be.
You can go anywhere your wizardly heart desires. If you're feeling generous, you can bring your friends with you. Teleporting is the ultimate in laziness. Did you need something on the other side of the city? Teleport there. Highly recommended is this glyph: Glyph of Rapid Teleportation. It gives you a 70% increase to your movement speed for one minute after every time that you teleport. You can use this to your advantage to sprint around wherever you've happened to teleport.
Did you know that there's a portal/teleport spell that isn't learned from a trainer? It's found in Scarlet Halls in the form of Ancient Teleport: Dalaran. Once you've looted and gleaned the knowledge from this book, you'll be able to teleport yourself accidentally to the wrong Dalaran (thank you, Slow Fall) or even better, teleport a friend! It's possible I once did this to a colleague, but you have no proof.
The trick to obtaining the book is to kill the last boss, Flameweaver Koegler, in either normal or heroic Scarlet Halls. You must intercept each of his Book Burner casts, though, or he'll scorch the teleport book you're after, along with some others. Once you've successfully blocked Book Burner and killed him, you have to make sure to loot his body. Now that you've done that, if you stand in the middle of the room facing the entrance, you should be able to see a blue lootable book on the shelves. It looks like this.
You can loot the book at level 85, but you won't be able to learn it until level 90. Be warned, though, that once you learn that teleport there's no going back and you'll have to be careful about which Dalaran you're teleporting to. Also note that Endora Moorehead in Dalaran sells this book for 150 gold (in limited quantity). You can find her in Sisters Sorcerous, over by the Alliance portal area.
Other portal shenanigans are well-known to all mages, specifically a rousing game of Portal Roulette. If you've never seen Portal Roulette, it works like this: you stand and face in one direction and then cast all of your portal spells one on top of the other. At the end, you'll have a circular light show that will send any hapless traveler to a mystery location. Who knows where you'll end up? But it's probably the Exodar.
Custom Polymorph Spells
Why stick to boring old sheep when you have an array of wildlife at your command? You could make it your mission to learn every available polymorph spell, or just choose the one that's most appealing to you.
Learned from Tome of Polymorph: Turtle. This is my signature polymorph. Formerly a rare drop from Gahz'ranka in Zul'gurub, it's not BoP and can sometimes be found for sale on the auction house. Otherwise, the only way to get your hands on one is to fish it up from Cataclysm zones.
This polymorph is sold, again by Endora Moorehead in Dalaran at Sisters Sorcerous. This book will set you back 2500 gold (minus any reputation bonus you have with Kirin Tor). I find for actual CC purposes this polymorph isn't great because the cat is so small that your tanks often won't even see it. It's very thematic during Hallow's End, though.
Learned from Tome of Polymorph: Rabbit, purchasable during Noblegarden for only 100 Noblegarden chocolates.
You used to learn this polymorph via a quest chain in Azshara. Now, it's trained from an NPC. Find The Amazing Zanzo in Dalaran and he should be able to teach it to you.
Bear Cub, Monkey, Penguin, Porcupine
These next four polymorphs are learned via glyph: Glyph of the Bear Cub, Glyph of the Monkey, Glyph of the Penguin and Glyph of the Porcupine.
It is possible to inscribe three out of four of these glyphs at the same time, and then using Polymorph just chooses one of the three at random. This also appears to invalidate your other known polymorphs (Turtle, Rabbit, Pig) instead replacing them with one of the three glyphs. If you only inscribe one of the glyphs, your other polymorph spells function as normal.
Now, you might be saying, what's the point of all of these polymorphs? Nobody CCs anymore! Well, that's not entirely true, but more importantly that brings us to Glyph of Crittermorph. This is a brilliant glyph that gives you the ability to polymorph critters. More than that, the critters you polymorph will stay polymorphed for an entire 24 hour period. This has plenty of possibilities just on its own. Polymorph every critter you see in a raid. Comb through an entire zone and polymorph the critters there. This is especially great in the Vale of Eternal Blossoms, where polymorphed moths turn into turtles that float through the sky.
But wait. It's possible to make this glyph even more fun. In the Vale, there's a rare named Moldo One-Eye. You can find him walking along the shore of what used to be Whitepetal Lake and the river there. He sometimes drops this item: Panflute of Pandaria. It makes nearby critters follow you for fifteen minutes and can be used once per minute. So if you have this, you can create a raid full of pigs and then make them follow you. Or fill the Vale with a random menagerie of creatures, or do your best Bo Peep impression.
There are even more entertaining glyphs for mages. Use Glyph of Illusion to borrow someone else's appearance for awhile. If you are on an RP server (the only place I think this would be relevant) use Glyph of Arcane Language to understand what your faction members are saying in their racial languages.
Use Glyph of Conjure Familiar to summon a little elemental of a random type (arcane, icy, or fire, naturally). If you're a frost mage, have fun with the size of your elemental using Glyph of Evaporation or Glyph of Condensation. Have a tiny icy familiar following you along with a little version of your elemental!
This isn't specifically for mages, but we are extremely fashionable and have some of the best tier sets in the game. I'm sure we'll talk more about transmog in greater depth some other time, but for now let's keep it simple: It's really fun to plan and organize the pieces for a transmog set. Get creative with it: What would your fire/arcane/frost mage wear? Choose one iconic piece and design a set based around it. Make a transmog to match your mount, or your pet, or a raid instance or zone. Make a set for the holidays! The limit is really only what you can get your hands on.
This definitely falls under the category of "completely random and frivolous things to do," but sometimes those are the most fun. Find the highest spot you can. Slow Fall yourself and then hop on a ground mount. Jump off and see how far you can get. Find a mage friend and race them. Be prepared to dismount to refresh your Slow Fall, since it won't last the entire time. I've made it to past Halfhill from the Vale doing this. It's actually a useful skill at lower levels when flying isn't an option.
So that's just a sampling of fun things you can do as a mage, either alt or otherwise! There are more I didn't have space to list, and I'm sure you've all got some I haven't even thought of. Tell us about them in the comments!