As an anivia player from long, this is a char who will b focused from mid to end game if you do it right.
UYsing anivia you'll find that taking down minions is so fast and easy, so you'll get lot of money even with no kills to get your gear up, so waste some money in wards for mid & things like that(oracle's, wards, etc)
It's an amazing carrier, and with the eggy time, you can save some ***es. The point is that you'r not a front char, so stay back all the time and help with that stun + huge damage on the teamfights.
As the carrier we are, we just need AP AP AP AP so ap runes + magic penetration it's the perfect choice, but you can try also some mp recover, if you'r not ok with the catalyst at first.
The quint, may vary from +5ap or +2 magic penetration, but on low lvl, your dmg will be low, so try to just reach lvl6 on mid and it should works.
Magic pentration marks are the best choice for sure, and on seals, you can take mp recover or ap/lvl, any will work fine for anivia. The ap/lvl glyphs are so cool, so this is not an option, it's a must.
Any mage masteries should work, focusing on cooldown and mp, so you'll be nice with that materies/runes choice. As I use clarity as spell, it's always helpfully to get your friend's mp recovered so a point there it's not a waste.
Exp mastery it's a completely waste, because you can stay on mid perfectly, as your oponent don't if it goes on the right way. Just a few chars can kick your *** on mid, and in that case, stay on tower and watch minions die on your ultimate, while someone comes to gank.
Starting mp will helps a lot to throw some balls here and there, and the 2 hp pots mayb needed while you take care what you get in front, so dun waste them, and learn your opponents moves before go in front of him.
Catalyst is a must, coz mp+hp recover is amazing on a low hp as anivia, and with catalyst+egg you'll b perfectly while you let your minions take down the tower.
Once catalyst is done, you must check how u are, how low hp your mid oponent is, and then focus on meja's or boots first, mayb with 2-3 lucky kills on mid you preffer to take a fast RoA, idc but dun be so hurried, or you'll loose the catalyst benefit. If game isn't going on your favor, focus on RoA + Rylai and forget meja's as mayb you'r the only positive on the team, and your bro's are feeding a lot.
If u got meja's and you'r owning, take rylai after the RoA, so your charges last longer, and u hurt a lot more. If meja's is not a choice, 2 archangel must be on the final build.
1st Q = flash frost = nice damage + stun damage. This means that your Q hits on the way, and when you xplode it, pressing Q again. The stun comes from xplosion, not on the way, so try to xplode your skill after it hit's a champion, to get double dmg. Won't take more than 3-4 games to get used to it.
2nd W = wall, nothing to xplain, a simple wall that helps to gank/run away. Use it properly on low lvls and you can block a complete way in the jungle.
3rd E = frostbite = main damage source. Using it alone, it's not an option, so dun waste your precious mp on this if you dun hitted with Q or with ultimate. If you do so, you'll get double damage on the spell, that is huge, on low AP start (lvl3 or so) and killing when you got your ap raised a lot.
4th = R = glacial storm = ultimate. This is your minion killing font. Use it always to get down a full wave of minion, after hitting them with ff (Q) Place it on the teamfights, on taunts, or even to scape, it's relly cool dmg + slow + low cooldown so just enjoy it a lot!
Passive = Egg = You'll have a second try in 6 seconds, but probably u'll b dead by that time.
You can try to chase a bit on mid when you got egg active. It was seen many times that minions complete the kill on the mid, while the enmy tries to kill you with that low hp you leaved him hehe.
If u get meja's, never run for a kill with no egg, u'll get the kill, but u'll die 75% times
As you got huge damage when you hit with q+e I focus on that, and forgive the wall till I have no other choice hehe, but some ppl thinks it helps a lot idc. When you get used, you will kill, and no run.
An important point is that you must hit with flashfrost to throw the frostbite, if you didn't hit+stuned (to double dmg 1 spell) don't waste your mp, keep moving around waiting cooldown to try again with ff first, if not, bite it's a waste, unless u got ulti, so u can start with that too.
My choice is always q+e+r on the way back, coz yes, they'll run back if u got them with that 2 double dmg spells.
Important tip is when you versus a teleporting enemy, like Ez, or Ktarina, don't start with ff (Q) try to get them first with ulti(R) so you can stun them when they port throught the ultimate. If you don't do so, they will port on the ff way, avoid it, and hurt you a lot, as you'r completely paper with no stun.
Clariry will help a lot to get on the catalyst, using it 2 times, and it's cool to help your team with mp, when they run out of it.
Ghost is an evasive spell for me, but also works as flash, but dun help to move on the lane when you need to. Remember that you'r the slowest bird ever born.
I don't like ignite, or any atack spell, just because u don't need it, and your mp/life is a thing to take care of.
Some people like teleport as an evasive spell, using it when you'r dieing, finishing the way while you'r an egg. I'm not sure if it was already changed, but I think nope, so you can try this also. It's like a so fast ghost hehe, but don't help to chase or run to teamfights when no minions :(
As a resume of the guide, focus on getting your ap higher, dun waste mp throwing fb if you didn't hit with ultimate or ff, keep always your work on the back, and push asap the mid tower, as you can go help any other lane or keep going to the second one.
Don't be scare to use ghost to run back to your tower, or helping some ganked guys.
Also it's not said yet but anivia is a HUGE FARMER! so exterminate minions any way you go, to keep your gear updated even with no kills. On a normal game, u must get catalyst on lvl 5 with no kills, and using ghost to run back to mid, you will throw a ff, lvl up and start eating minions with your ultimate, stop it to the car minion, and it's done! you got all your minions otw to the tower hehe.
Have fun, and make any reply if needed.
So sorry bout my english, but I'm not practised on this.