As Dragonrage pointed out in his answer Annie can be played on pretty much any lane (even in jungle but I wouldn't recommend doing this in a serious game).
Annie's strongest lane depends on your personal playstyle but keep in mind that statistically her strongest lane is mid-lane. I'm only covering Mid and Support here since the other lanes are purely situational.
Midlane is by far Annie's most played lane and as of writing this question she has the 3rd-highest winrate of any midlane champion. One of the reasons she's extremely strong mid is her Q. If you last hit a minion with your Q you will get your mana back and the cooldown is reduced to 2 seconds instead of 4. This not only allows you to farm easily, it also builds stacks of your passive which can put a lot of pressure on your opponent (especially if they have no vision on your jungler).
Not only is her laning phase relatively strong, she also has a lot of snowballing potential and can set up ganks easily. Her passive stun also makes her great at ganks and roaming in general and midlane is the best lane to play a roamer since you can help your jungler, go bot or top.
Her only weakness is her lack of a gapcloser and once you burn your flash you might be very vulnerable against strong mid-gankers like Shaco or Lee-Sin.
As core items there are two options when playing mid: Full snowball AP with a Morello/Hextech Protobelt start or a slower scaling but safer Rod of Ages start.
Botlane Annie isn't as strong as she used to be. She's played as either an AP or Tank support but there are quite a few weaknesses to her kit on bot
- She is relatively low-range compared to other supporters
- She is extremely squishy and will almost certainly die if she gets caught by CC (which makes counterpicking her easy)
- She can get some serious mana-issues when playing aggressive since you shouldn't be stacking your stun by last hitting minons.
- There are simply much stronger picks right now. Zyra does basically the same with delayed cc but more damage and range and other AP supporters have either more CC or more Heals/utility.
That being said she can still be a viable pick, especially on lower rankings. Support Annie is amongst the top 3 strongest supporters when it comes to level 6 burst. As soon as you hit 6 you can basically oneshot an ADC all by yourself. Up until then however you have to play carefully. It's still extremely important that you poke your enemies with auto attacks and spells but depending on the enemy support an error in positioning can mean a free kill for them since you are so fragile.
Keep in mind that support Annie will never come close to the gold of a midlane one. So your go-to build is Redemption into full-tank, effectively making you a stun-bot. Now if you are ahead you can risk it and go for a more AP heavy build but one slot should always be occupied by sightstone.
Now as I've already mentioned you might be a better support than mid player (or the other way around) which makes Annie automatically stronger in that lane (at least for you). Midlane and support have completely different carry playstyles and if you like her bot the most, go ahead and play her! It's certainly possible to reach a high rank with her on botlane. But keep in mind that her overall potential is greatest in playing mid. (At least until Riot decides to throw the meta around again)