An alternative start can include Doran's Ring + 2 pots if you prefer the early mana for more aggressive trading. I start dark seal + refillable because it's the most cost-efficient start and helps me build into my core items (aka. Mejai's Soulstealer).
I almost always build Liandry's. The only time I build Luden's is if they are heavy AP - when they are nearly an all magic damage enemy team comp.
Item 3 is Mejai's Souistealer. If you're not comfortable building Mejai's, Zhonya's Hourglass or Banshee's Veil will suffice instead, depending on the enemy team comp. If you're not sure, Zhonya's is usually preferable at this later stage of the game.
Once you complete all items in the build, you can either choose to keep your boots (I usually do), or replace them with Zhonya's hourglass. If you do not have Mejai's, do not sell your boots. You will be too slow.

The runes I use are designed for increased DPS and mobility for improved kiting in-combat.
Going electrocute is an alternative that works, however I find that the increased mana / cdr / in-combat mobility from the comet-inspiration runes synchronize better with the sustained damage and kiting potential of the liandry's-rylai build.

What to do levels 1-5 depends largely on the matchup. In general, in mid lane you will face four different types of classes: Assassins, Mages, Fighters and Tanks. I outline the general strategy behind what to do in each of these cases:

Assassins:

Assassins tend to be difficult to lane against, even though they are generally of the melee class. Levels 1 and 2 generally allow Annie to trade favorably against them, as most assassins tend to try to all-in you rather than poke you and thus need all their abilities leveled to do so. So they would need to be level 3 to have one point in each ability. This isn't true for Talon, who will all-in you at level 2. So generally, Q and auto poke against assassins levels 1-2, and then post level 3 you need to make careful consideration that either:

a) Have your stun ready to begin the trade or
b) they have used abilities on the wave and will not be able to all-in you should you decide to step forward to trade.

Generally speaking, Annie is at her weakest point at level 5. Do not risk dying to an assassin at level 5 for a CS. Simply back off and wait for the wave to come back to you.

Mages:

Mages can be subdivided into three different subtypes. Melee, Short-range and Long-range.

Subtype melee:

These tends to be the easy-medium matchups. Sylaas, Ekko, Diana. Champions that tend to be tankier but again, you have no problem using your combo on and are generally easy to gank. Depending on who you're facing you shouldn't have any trouble farming to level 6 in these matchups.
Just be careful and poke only when they've used their abilities on the wave or are going up to CS a minion and you can get in an auto for free. Usually, you won't kill them unless they accidentally get too close to your tower. Ganks are always an option vs melee champions.

Subtype short-range:

These tend to be the easy matchups as well. Twisted Fate, Vladimir, Ryze, etc. They are easy because they tend to be squishy and do not outrange annie, and thus Annie has no problem using her combo on them. Still, you need to be careful about trading early as Annie tends to be weaker until later in the game. So generally trading when they are about to CS or when you have a stun is usually best.

Subtype long-range:

These tend to be really annoying. Azir, Viktor, Syndra, Neeko. Honetsly, I have found it best to build MR in these matchups. So going back and getting an early Verdant Barrier will do you wonders in these kind of matchups, as most of the harass you will receive is going to be next to impossible to avoid. So, banshees Veil is recommended as item 2 after mythic, with verdant barrier being bought early to survive the landing phase.

Fighters:

Generally, fighters are easy. With the exception of Irelia and Tryndamere. More on those two later. Other fighers include Yasuo, Yone, Wukong, etc. These matchups you can poke levels 1-5 and there's not a whole lot then can do. Try freezing the wave and setting up a gank with your jungle. They don't tend to do well when they're sitting near your tower this early as they have limited ways to escape.
You are also in a safer position to harass them as any time they try to trade with you your tower will aggro onto them.

Irelia:

Be very careful with irelia as she will shred you to pieces if she lands her E stun on you. So, early trading is fine, so long as you avoid the stun. Honestly, armguard isn't a bad option versus Irelia as it will mitigate her all-in potential greatly.

Tryndamere:

Again, like Irelia, if tryndamere gets on you you will be quickly destroyed, especially since Annie has no escape options and Tryndamere has that annoying slow. Zhonya's is pretty much a must in this matchup. So build that second, and then continue building towards rylai's as a third item instead of it being second. Armguard early will make it so he cannot dive you so easily after he reaches level 6.

Tanks:

Tanks tend to be easy. Champions like Ornn, Mundo, Chogath. Usually, tanks are squishy early and are very gankable with your jungle as they are melee. You just have to avoid getting into melee range with them. So avoiding skillshots and not overstepping into their range is to be preferred. These matchups I consider going tear --> archangel's as the extra mana can help versus the HP and MR they inevitably build, but this isn't necessary. First strike is also a good option versus tanks as you generally get to hit them first before they hit you.

Once you have ultimate, you have significantly more kill pressure. Setting up a gank by pulling your wave and freezing it is usually best until you have your tier 2 boots built. Try not to roam too much if you don't have boots. You're too slow and you will miss a lot of CS. So only roam without your boots if it's convenient, aka:

a) the skirmish is close-by
b) the enemy is greatly overextended in a sidelane
c) your team is doing an important objective like dragon or rift herald.

Once you have your boots, it becomes much easier to roam. You can look to set-up plays with your jungle after you shove your wave in middle, or, if you notice a large ally wave building up in a sidelane, you can walk to that lane and dive with your laner with all the minions. These types of roams are brutal for the enemy laner as not only do they die, but they also miss a massive amount of gold and XP from the missed minions. So, keep in mind the status of the minion waves in sidelines to decide which way you want to roam before you commit to something. This is an important aspect to consider if you want your roams to be successful.

Teamfighting can be tricky. Often times it is simply a bunch of random champions dashing forward and backward, in and out, skillshots going everywhere and in all the chaos you find yourself trying to decide the best course of action in a small amount of time. Generally, there are three main approaches to teamfighting. These three approaches are as follows:

Focusing the Frontline

This approach means, hit whatever is in front of you and don't go deep into the enemy's position. This is usually my preferred method of teamfighting, as I tend to build Liandry's and so am more able to shred frontline tanks or any carries that get caught up alongside them. But this approach also has its cavest. In general, focusing the frontline is best if:

a) The enemy team has little MR. If the enemy team doesn't have much magic resist built, you should focus whatever is in front. Going deep into the position against a team of squishy carries will just get you killed very rapidly. So play with your adc here - play to kill whatever is near your or focusing your adc.

b) Eenmy team is a bunch of fighters / tanks / melee. Just hit the frontline here and you will have served your purpose. Do not try to go past the sion to focus a yasuo or yone. Just let them try to come to you instead as you kite backward.

Focusing the Backline

When is it appropriate to ignore the frontline and instead, try to use your damage for the backline, then? In what cases should we ignore a tank and instead go for the support/carry heroes? In general, there are two cases where focusing the backline will be preferred.

a) The enemy frontline has heavy MR and you have flash available. So imagine a Sion with a spirit visage, mercs, and force of nature. It will not be beneficial to focus him as the amount of damage you will deal to him will be minimal. It is better in this case to focus a squishy carry in the backline if you can reach them with flash. If you are successful in killing a carry, backoff and play to protect your adc.
b) The enemy team carries have clumped together and you have flash. This is the best case scenario if you do have option. Killing one (or two) squishy champs in the opening fight will surely net your team the victory in any teamfight.

Protecting your ADC

So, what happens if you don't have flash available, and you feel like hitting the frontline super MR tanks will be a waste of valuable ability resources? In this particular scenario, you should try to protect your adc and let them deal the DPS instead. This strategy is particularly useful if the enemy team is easily kited. The rylai's slow will help your adc kite the enemy and make it easier for them to deal the damage necessary to carry the fight.