Read the guides for Anivia that are posted, but what you really need is a few more practical things.
1) Get someone to play as Anivia vs. you. A lot (practice games with 1v1 are good). You'll see how you, yourself, counter her and then you'll gain insight as to how people will approach you and what people think when you're moving about and such.
2) Reading guides does NOT make you a better player. It puts you in the right mindset and you know what you need to do, but accomplishing it is completely different. Make sure you really feel out spell combos and, more importantly, when to run and when to fight.
3) Make friends with fellow Anivia players and play practice games WITH them. Have them play Anivia, and you lane WITH them, even if it's just with bots. You'll see what to do, when to do it, and how to cooperate with people. Then have them lane with YOU as Anivia, and do what you can together.
4) Decide on a rune page for Anivia. Personally, I go flat HP on quints, MP per lv on Marks and Glyphs (+350mp by 18 is excellent, and really bumps up the power of an archangel's staff as well) and HP per lv with seals. When I get more IP to try out several other builds I'll probably find some hybrid build, but for now, MP is ALWAYS useful for Anivia in one form or another, and you can never go wrong with more HP for her survivability and egg advantage.
For early game - Lots of people say to get a saph crystal and HP pots, but I find this to be incredibly wasteful. I use an HP crystal, and with my runes, I'm pushing 670 HP just to start, which will keep you alive much longer. I also use ignite, and with that, I can be REALLY aggressive and get tons of early kills and push to my catalyst and boots by lv 5-6 easily. Also, to curve the MP problem, use Clarity and you'll do fine early on, just make sure you only boost FF when necessary and cram FB as fast as possible. This will cause you to limit usage of MP because FF (80mp start) will get you to your chilled status for FB, which costs MUCH less MP to use and you can get the same amount of damage out of it.
And then I usually max out FB ASAP, followed by FF to 4 (usually 5 though) and then I get my wall. Though, wall lv 1 or 2 is sometimes really good for blocking jungle lanes
and keeps FF mana usage down if you decide to get that before maxing FF (which increases 20mp per level, as opposed to FB's 10mp per lv). FF maxes at 160mp, FB maxes at 90, it's pretty clear that FB does the same damage minus the stun, for gobs less mana. Also, if you are low on mana, just toggle storm quickly and then use FB, it's nice damage and minimalizes mana to ~180 or so, rather than FF+FB's 250, but that's also situationally dependent.
I simply cannot express enough, Anivia is an early game dominator. Pressure your lanes, learn to zone control (search for Shurelia's video on zoning techniques on the forums or on youtube, it's INCREDIBLY helpful). Anivia is fabulous at controlling her lane, and if you force lane control, your opponents will learn to fear you.
As far as item builds, there's a few that I use depending on the situations my team is faced with early on.
- If we're down kills, I go tank mode, which is RoA, Merc Treads, Warmog's, Banshee's Veil, then Void Staff if necessary (though that's a long game if you get that far).
- If we're even on kills, I'll go half tank and get RoA, Sorc/Merc (depending on how you've felt out the battle and what you think you need), Banshee's, then Arch (banshee's mana boosts arch's AP gain, since it's based on your max mana pool, and it's quite a nice item for a half-tank build).
- If we're steaming with kills, I go RoA, Sorc Boots, Arch, then Void Staff. This keeps the opponents afraid of you and they most likely run, rather than fight, and this is especially helpful in teamfights for quick burst damage on the biggest offender. Anything beyond that for each build is usually up to your discretion, but more HP is doubly helpful for Anivia because of her egg.
Check out the notes in that thread and watch whichever of the vids you choose to tolerate. It might help you out with some stuff, and you might pick up some techniques you never really thought about before.
My last piece of advice is to never give up. I never really had any champion I truly liked until I found Anivia. I previously played Ashe and Annie, and I played one round with Twitch, but once I got hold of Anivia, I never looked back. I've played 120+ games with her since and I'll never pick another champion. The neat thing with Anivia is, there are SO many different ways to accomplish things with her, and I think that's the greatest thing about her; you'll always discover something new, something else to add to your arsenal. The only hard part is remembering all of them and recognizing when to use what technique, XD
If you want any help, don't hesitate to add me to your friends list and we can help you get some more experience under your belt.
Edit - Another thing with Ignite I forgot to mention, it has a larger range than FB, so if you get in close enough for FB, you're already in Ignite range. I love this and use it to my advantage all the time, it can really bump your damage limits up by a ton in early game. By level 6 you're doing an additional 200 damage, which is about equal to the amount you're doing with FB (champs have some magic resist so you're never really doing the complete 115x2 damage). FF + FB + Ignite is easily 600 damage, so if you can pull that off, plus storm damage, you can get 8-900 damage in with auto attacks in no more than 3-4 seconds. Most champs don't have more than 1200 or so by this point, so it's quite a large burst and can help you get quick, decisive kills early game.