Catalyst the Protector, Clarity, and Teleport work together to keep you out there farming and full of mana for a long time while you level. This is a huge farming advantage, as Anivia has a great minion-slaying skill, but one that is usually too expensive to be used for this purpose.

I have two builds here - one's a vanilla AP build with some magic penetration, and one is a support/debuff build that's also great for pushing turrets. The latter seems more interesting and useful to me, and the former is probably easier to learn and "mage-ier".

It's true that Anivia is a little tricky to play. There are lots of guides that cover skill combos and such, which is valid for just about any build. I'll just review a few points that stand out here.

* Note that you can hit Q to detonate Flash Frost early. This is important - you want the rock to hit your foe, and *then* you want the detonation to do additional damage and stun them. The damage figures in the skill descriptions are misleading - Flash Frost can hit twice, and Frostbite does double damage if your enemy is hit by Flash Frost or Glacial Storm first.

* Flash Frost is a skill shot, so it can be tricky. It's easier to land at close range, and you can hit AI from even further because they don't have any concept of dodging.

* Glacial Storm can be deactivated after casting... and it should be. Hit R to stop it as soon as you don't need it anymore.

* Crystallize is tricky to use right, even for experienced players (if they're new to Anivia). It's not that useful early in the game, so it's ok to skip it altogether until level 7 or so. Hitting harder with Frostbite may be more useful.

Q = Flash Frost
W = Crystallize
E = Frostbite
R = Glacial Storm

AD = Attack damage
AP = Ability power
CC = Crowd control. Stun, silence, fear, taunt, etc.
Debuff: Temporarily lowered armor, MRes, healing rates, etc.
ASpd = Attack speed
MRes = Magic resist
MPen = Magic penetration (allows *your attacks* to bypass this amount of MRes; only you do extra magic damage)
MRed = Magic reduction (temporary debuff lowering MRes; any ally attacking this target do extra magic damage)
MP5 = Mana per 5 seconds (mana regen)
OOM = Out of mana
DPS = Damage per second. Either from right-click, or from skills that have very short recharge time.
Nuke: Large damage and/or over a wide area. Either costs a lot of mana, or has large recharge time.

This is the key feature of *both* builds I give above, what they have in common. The plan is to mana dump and farm hard early. Some of the advice I will give here would appall experienced players, but the choice of summoner spells leaves room for mana sloppiness. As a result, we get a build where you can make "noob mistakes" and still do alright.

You may want to buy a Mana Potion or two with spare change from time to time, but if you get good, you can do without them. It never helps to use two or more at once; all it does is reset the timer. It's better to use Clarity first, because then you can use Clarity again that much sooner.

If you're a total noob, it's better to err on the side of carrying a couple health and mana potions

. Making a mistake and taking damage or using too much mana or whatever can cause you to leave your lane too soon. It's not great to lose hundreds of gold doing this, but it's better than losing exp and gold by making more trips.

Levels 1-5

Try to get a shared lane, to start with. I like to roam after level 6 with this build.

This will be the hardest. You want to get enough money for Catalyst the Protector. Generally, any time you can pull off your QE combo on a champ from a safe place, do it. Run out of mana. Use Clarity, starting the cooldown timer. Spend more mana, pop a Mana Potion. Run out of mana again, Recall, shop, and Teleport right back. If you have a good first trip out, you *might* have enough money to complete the Catalyst, but this depends on your skill and your opponents' skills.

Levels 6-14 or so

Glacial Storm changes everything, especially if you have Catalyst the Protector. Get 6 or more enemy minions to pile up on you or one of your minions. Pop R under a bunch of them. Un-pop R when they die. Push minions all the way up to the turret. If there is a tight cluster of 3 minions, if you can last-hit two of them with Q, that's better than R or auto-attack.

If you're confident enough to try to pull off a gank, then you can roam to the middle. Sometimes, you can even Teleport to finish off some foes who are at a disadvantage. If you aren't out to gank, then you can at least suck up some minions and harass the enemy champ. You might have an easier time just staying in your lane, though... new minions spawn often enough that you can farm just one lane with R for pretty good gold.

If you have Catalyst, try not to use Clarity right before you level up. If you restore your mana *above* your max MP, then you lose mana.

As you get close to level 18, upgrade your Catalyst for a Rod of Ages.

Your right-click isn't good for much, but it's easier to spam spells. Pushing lanes is easier, but you can't take a turret down rapidly without an ally.

If you're doing well, you'll have ASpd bonuses after your 2nd or 3rd trip back to camp. This helps you take down turrets rapidly once your minions are on the doorstep. You can decide for yourself what order to get the last items in, but the issue of Nashor's Tooth vs Malady as your next item, I'd like to elaborate on.

Nashor's Tooth before Malady: Mana regen and CDR are nice. It's starting to take longer between levels, so if you aren't good at conserving mana between Clarity and Teleport cooldowns, this can help a little.

Malady before Nashor's Tooth: If teamfights are breaking out, then Malady will make you more useful as a support character. It can pay to find out who's doing the most magic damage and helping them out from time to time. You'll be slowing, stunning, and MRed'ing wherever you go, but your damage output isn't as great as a pure AP build. If you can use W effectively, then this can also help, but if you're getting panicky, then it's best to err on the side of not using it in the heat of the moment.

You may want to use R slightly less, since you can actually farm somewhat decently with right click. But if there's a ton and you want them gone quickly (say you're right on their doorstep with an ally and no enemy champs around, for example), go for it.

One last remark: if you have right-click DPS on your team, you might want to encourage them to pick up Madred's Bloodrazor - this does magic damage, and is subject to bonuses from MRed! If you can slow, stun, wall in, and get 4 auto-attacks (for Malady's unique passive) in on a foe, they won't be happy at all with what your DPS ends up doing to them.

Why no Void Staff?

It's a great item, but %MPen doesn't combine well with FLAT MPen. %MPen is applied AFTER flat MPen. If your foe's MRes starts as X, and is modified to be Y, and you have 20 flat reduction and 40% reduction, then Y = .6(X-20) = .6X - 12, meaning your Sorcerer's Shoes aren't helping as much. If you want Void Staff, get it fairly early and take Mercury's Treads or Ionian Boots of Lucidity instead of Sorcerer's Shoes.

The given build has you getting Abyssal Mask to pick up some of the slack.

What about Deathfire Grasp?

1) Not noob-friendly enough; it has an activated ability.
2) Also, it costs more than Morello's Evil Tome and has less CDR.
3) ...But if you want to do it, I could recommend Ionian Boots of Lucidity and Deathfire Grasp instead of Sorcerer's Shoes and Morello's Evil Tome. Or you can keep the shoes the same, if you're comfortable with the CDR; getting the blue buff from the ancient golem is a possibility.

These have mostly been discussed already. Will of the Ancients is iffy, but it's cheap and this late in the build, it's really nice to help your teammates stay in the fight. After you lower their MRes and slow/stun them, put this to good use by helping your team heal with 25% spell vamp. (Spell vampirism; it restores life when you deal magic damage).

If you're totally rich and kind of selfish, you could take a Rabadon's Deathcap at the end, but I wouldn't recommend it unless your DPS/nuke players are all incompetent and you have to play carry.

I'll consider alternate 6th items later.

Pros:

* (Relatively) noob-friendly
* Fast farming
* Good support/damage hybrid

Cons:

* You might pick up some bad habits
* Possibly not viable at high Elo, where other summoner spells used wisely may be more useful than tons of mana
* Squishy
* Not *that* noob-friendly

Option 1

Option 1 pros:
* Waste even more mana
* Get more kills
* Farm a little faster

Option 1 cons:
* Totally mana dependent
* Encourages even more noobish bad habits

Option 2

Option 2 pros:
* OOM? Right-click! It actually does something this time around.
* Team player
* Less MP5 encourages you to right-click when appropriate
* Nifty debuffs from items work well with CC skills

Option 2 cons:
* Depends on teammates more
* Less mana means less CC