Pros:

Her passive can save u ALLOT and save ur killing spree or can bait some 1 to tower dive u.
Her farming at lvl 6 is to gd.
Her burst is CRAZY with E.

Cons:

Really squishy early game.
Her Q animation is really slow can be so hard if u new with her to land ur combo (Q-E).
To slow ... SLOW* .
if u was to over extended ur passive well not be that useful so try to not be.

With Anivia i love to get mp and flat ap and armor runes .
Why?
Flat ap and mp give u the burst u want early game.
Armor is a good way to help u against ad mid Example: a gd cait mid can be a pain in the *** and aramor runed well help allot against her or any other ad.

Why i don't use mana runes ?
mana regain runes it's gd with anivia but i don't fined it that useful mid and late game (it can be solved by a blue buff and the item build)

Anivia might be hard early game especially at farming ,her animation is realy slow and am sure u know that .

With anivia try to avoid enemy harassment as much as u can and look at ur minions health so when they come to take it u just land an easy Q - E combo.

Here is an easy way to farm from lvl 1 to 5 , auto attack the caster minions and bring them like to half health then Q them (u need to het them twice with ur Q by re-activate it after it hit them).
From lvl 6 Q all the minions and ult ,NOTE: don't use ur ult and just leave it on for a long time wail farming just use it for caster minions and turn it off.

With Anivia i love to put a 2nd point in my wall (but not max it b4 Q) ,lvl 2 wall can almost block the path from the bot river to ur jungle if ur team is blue and from top river to ur jungle if ur team is purple ,it can be very useful.

Spells combo lvl 1-5 (Q-E) lvl 6 (R-E-Q-E).

*MID* always ward if they have a jungler , ward the bot brush as a blue team and top brush as purple , then u can over extend but be as near as possible to ur warded brush.

Here is a good strategy i almost use it every game:
(blue team) the jungle attempt to gank u from top river and ur mid enemy about half health same as u ,run to bot river and go to ur jungle the jungler well chase u and the other enemy well attempt to cut u off by going the other way , when u reach the path to ur jungle block the jungler by ur (lvl 2) wall the mid enemy well be in ur face if he tried to cut ur way ,don't panic and fight hem it's 1 v1 u can kill hem in seconds even b4 ur wall times up.

Late game when u die in team fight u well give the enemy 2 options 1- they aim u in egg form and ur team kill them all 2- leave u and when u bk u kill them more.

Don't place a wall if u not sure about it ,the wall can kill u or ur team.

If the enemy team was taking baron try to stop them with ur team ,when any 1 of them go out from baron place a wall behind hem and trap hes team into the baron place so it's 1 or 2 v5 , now baron is useless for them cuz they got aced from ur wall .

Another thing if u have a ward inside baron u can steal it by going behind the wall an click on it to see baron hp ,when baron reach 900 hp us ur q it gonna hit baron twice 300 each if u did it r8.
NOTE: 300 each depending on the game time and what u building and 900 hp cuz Anivia q is realy slow until it reach baron well be about 600-500 hp and u need to hit it TWICE.

For any ap caster i like to build tanky ap , like any item that gives health/armor/mr and ap .

Mana regain items is a mian thing with Anivia she's mana *****.

My build well make u very tanky late game if they aim u u can easly use ur zhonya's to confuse them and still u wont die easily cuz the hp/armor/mr u have ,in the same time u still gonna do a CRAZY dammeg to all of them.