Keep an eye out for a white, swirling power around Annie. It means she's ready to unleash her stun.
Disintegrate is extremely useful in lane. Even at low ranks, its cooldown is extremely short if it kills its target. Along with the mana refund, this makes it an excellent farming tool. This also allows Annie to build up her stun quickly and freely as long as every cast of Disintegrate kills something.
Disintegrate is a good choice to max first in a lot of situations. A higher-level Disintegrate means more reliable farming and increased damage on a targeted ability, while only increasing its mana cost by 5 per rank, which can be almost entirely negated by the mana refund on kill.
Incinerate should be taken at first level instead of Disintegrate if you need a stun at level 1 or if you're playing support. The main reason is that it can be cast without a target, meaning that Annie can build up her stun in base to negate the mana cost.
Molten Shield has two main uses. The first is to give Annie a decent defensive buff, which can help her to win trades in lane or survive teamfights long enough to dish out her burst. The second use is as a cheap charging tool for Pyromania, thanks to its low mana cost.
It is very rare to see Molten Shield maxed first. While it is more viable than the old Molten Shield due to being a percent reduction of damage instead of a flat bonus to armor and magic resist, the benefits of maxing Molten Shield are negligible in comparison to her other basic abilities.
Partnering Summon: Tibbers with Flash leads to a near-instant stun with surprisingly long range. However, try not to be too obvious when preparing this combo, as a cautious enemy will probably back off if they anticipate an engage.
Don't feel like a fully-charged Pyromania is too valuable to waste on farming. While most enemies won't come near you when you have your stun ready, keeping it for too long might mean you miss out on farm because you don't want to use Disintegrate. Given how quickly Annie can build up stuns when last-hitting with Disintegrate, your stun won't be down long enough for the enemy to fully engage on you unless you're too far from your tower.
Doran's Ring is a great starting item on Annie, as she can make effective use of all the stats it gives. The extra health especially is very useful during trades due to her short range, and the on-kill passive has fantastic synergy with the mana refund on Disintegrate.
The Dark Seal is dangerous to start with, but it can pay off if you're able to keep yourself alive or if your teammates can protect you well. However, you should never start with this item if you aren't completely confident that you can win lane without dying once.
The advanced items you buy in lane can vary a lot, but there are several notable options that see a lot of use.
Catalyst of Aeons provides health and mana, making it very useful if you find yourself running out of either during the laning phase. In addition, its passive has great synergy with Annie: her low range means she takes a lot of damage during trades, which will give her mana back with a Catalyst, and the refund on Disintegrate means that you're effectively healed for free whenever you kill something with it.
Blasting Wand and Needlessly Large Rod are decent buys if you've killed your opponent or forced them to recall at least once. In these cases, you've probably got a fairly large advantage already, and the raw ability power given by these items can push you far enough ahead of your opponent that they will be unable to make any form of comeback in lane. However, you need to be very careful if you choose these items, as one well-timed enemy gank can easily kill you given your fragility and lack of sustain.
Finally, Fiendish Codex is similar to the above two items in that it provides a decent amount of raw ability power and can be used to close out your lane. However, while the cooldown reduction is a useful bonus and can make you even more oppressive towards your opponent, you have to be very careful to watch your mana to make sure it doesn't run out at the wrong time.
Your first complete item will obviously depend on how your lane went and what you've already bought, but you should always consider building Morellonomicon first. The ability power, mana, cooldown reduction and mana restore on kills and assists make it a fantastic item for Annie, and the passive's Grievous Wounds essentially makes it impossible for your target to be saved by any form of healing or lifesteal.
Unless you've got a large advantage in gold and levels over your enemies, Rabadon's Deathcap is not an ideal item to rush first. While it does provide a huge amount of ability power, it has nothing to back up that power with. You'll often be better off going for cooldown reduction or sustain on your first item.
If you find yourself being focused, buying a Rod of Ages or Rylai's Crystal Scepter can help to prevent you being bursted down before you can do anything. Both items provide a fair amount of health, while the former provides a lot of mana to keep you alive and able to cast spells and the latter will slow enemies that try to chase you, which is also useful for catching your own targets.
While it is listed as an essential item, Luden's Echo doesn't need to be bought first, second or even third. Like Rabadon's Deathcap, it gives a lot of burst and very little else, though the movement speed might help you get away or catch enemies.
Like Rylai's, the passive of Luden's Echo triggers on dealing damage in any way with any of Annie's spells. Be careful that you don't waste it on a minion wave because Tibbers walked up to it or you were taking minion aggro with Molten Shield active.
Because of Annie's short range, Abyssal Mask can be a good buy, especially against another mage. However, a Void Staff might be preferable if you want straight damage.
Zhonya's Hourglass is good on Annie, especially against champions with massive burst and/or high amounts of physical damage. Notably, if Tibbers becomes enraged less than half a second before you go invulnerable, his rage can last for your whole invulnerability period, allowing you to inflict significant damage while in stasis. In addition, Tibbers can still be commanded during this time.
All of Annie's spells deal magic damage, therefore any form of magic resistance will mitigate her damage significantly.
Banshee's Veil offers a spell shield to negate 1 spell from Annie. Its magic resistance and health will help tremendously in surviving Annie's burst, while possibly blocking Annie's initiation. As such, it is a recommended buy if going against Annie.
Quicksilver Sash is a cheaper alternative that allows you to cleanse her stun specifically instead of just one ability; however, it does not prevent the damage.
If you're a support trying to protect your carries from being stunned, consider Mikael's Crucible to help debuff.
Do not underestimate the damaging aura from Annie'sSummon: Tibbers. The longer the fight lasts, the more damage you will take from it.
Avoid focusing Tibbers when attacking Annie, as Tibbers has a lot of health and a decent amount of damage. Focusing Tibbers allows Annie to finish you off with her spells if she is ignored.
Do not disregard Tibbers completely, however. Its attack damage and damaging aura can easily put a tense fight against Annie in her favor if left to attack you. Furthermore, an enraged Tibbers becomes very dangerous when you are stunned.
Alternatively, use ranged attacks and abilities to kill Tibbers while keeping out of range of Annie's spells, the advantage to this being that Tibbers will most likely be easier to kill.
Ideally, Tibbers should be taken out while Annie is under crowd control. That way, he won't become enraged due to Annie's death, and Annie can't deal damage to you while you kill him.
Despite Annie's powerful stun and burst in spells, Annie's spells are short-ranged. As such, long range mages such as Lux, Morgana, Ziggs, and Xerath, and long range marksmen like Caitlyn, Kog'Maw or Tristana can easily zone Annie out and prevent her from using her spells effectively.
If there is an Annie on the enemy team, avoid being clumped thogether with your team. Some Annie players are known for making unexpected Flashes when they have their stun ready, casting Summon: Tibbers and stunning your entire team. This can be potentially game-breaking, especially if the stun affects your allied carries.
Molten Shield acts similarly to Thornmail. Be careful when engaging her if you are using a champion that relies on auto-attacks.
Forcing Annie to use her burst combo on a frontline tank will stop her from immediately taking out squishy targets. In particular, champions like Alistar who can survive her burst and then lock her down will make Annie far easier to deal with.