I know a lot of people don't ward and say its the support's job. Well, playing support anivia is not an exception. Use those wards. Drop a pink in one of those bushes that people don't always walk though, but still provide a lot of traffic. Mainly, the ramp into the jungles around the 4 sides of mid.
Ward dragon, and make sure you have timestamps enabled so you can keep track of buff/dragon/baron timers. (5,6,7 minutes to respawn). Ping the spots to let your teamamtes know where enemies have placed wards.
Refer to one of million warding guides on mobafire. Drop a ward, save a life.
Picking your Items.
If high poke lane, consider starting doran's shield. If confident, run ancient coin.
If against squishy low attack range opponents, vayne/leona run spelltheif's edge to harass in lane for extra money.
When the enemy starts getting bigger items, switch your spelltheif blade into a coin since trading with AAs will probably start hurting.
Upgrade to Talism of Ascension when team begins roaming as 5. Using the active then using your wall can catch unsuspecting enemies.
Run boots of mobility to roam to mid and gank. Run sorc boots if ahead, extra damage as another APC is really good.
Tabi / MercTreds if required.
Rush liandry's since it alows for good damage with lack of items.
Next item is survivability.
Build Sightstone if you need hp. Hourglass to bait skillshots. If sightstone is left out, be sure to keep buying wards.
Build damage / survivability / actives depending on your team's needs.
Tips and Tricks
Auto attack your opponents for extra money. You should have little resistance due to your superior AA range. Only support Annie can fight back with another AA.
Your AA damage will stick since you have hybrid pen runes and AD runes
Try not to use your abilities unless they are overextending really badly.
When you have your ultimate, you can throw your E then cast your ultimate while the E is travelling to your opponent. This ensures you are actually in range to harass with your E and not just do 70 damage with your ultimate by using 100 mana.
Try to aim your walls between towers since the tower will mean they have to run farther around to get past your wall.
Your wall will naturally push opponents away when chasing. So if they are on the edge of your wall range, don't cast it, since you most likely will block your teammates instead.
In between CC casts, your wall and stun, make sure to autoattack. The damage from AAs must not be overlooked as an extra 54 damage can determine the outcome of botlane.
During early phases of the game, you can bait a double kill if your jungler is around by having them overcommit to egg you. Done right, your jungler and ADC would kill them both and you should revive from your passive.
Fighting as a TEAM
Your jobs in team fights is either damage dealer / pseudo-initiator or peeler.
Most likely, you will be peeling. When the fight starts, make sure to stand in front of your carries and block the assassins trying to jump your back line. Help your carries cut down their assassins and proceed to destroy the rest of their team.