First tip for basically any mid mage, FARM.
Brand is no exception. The reason behind ignoring Sear until Level 7 is to allow you to farm up faster, The only reason for you to pick up sear before your ult is if you are in fear or getting ganked.
At level 1 and 2 do not waste mana trying to farm and show you opponent some who is boss by throwing a few pillar's under the caster minions to scare them off from last hitting. with level two if they backed off the would think you got sear. this is when you use your E on an ablaze minion and catch them with some unexpected damage.

level 4 and 5 do the same thing with the spells only this time watch out for the caster minions and make sure to last hit if you miss the timing on the kills of them. last hit the melee minions as normal. you will push alittle with this but it is harder for you opponent to last hit under tower then it is in the open. just don't push too hard and it will be hard for junglers to gank you.

Level 7 you should have your full skill set to at least one point and with boots doran's ring and an needlessly large rod. with this last just one W with take out the caster minions. using your E to ablaze the target if you miss with your W then fulling it up with Q and and then R using ignite to finish off anyone who lives this burst.

You should still not ignore farming and should have over or becoming close to 100 farm by the 17:45 marker. (around the same time your carry should also have the same unless death says otherwise).

On brand Farm is your best friend, Kills are your reward.

When fighting in closed off areas like in the jungle or by dragon or baron. Remember to try and hit as many targets with W as possible then hit the target in the center closet to all your enemy's with your E, Save your Q for either
1. Chasing fleeing foes from a won fight.
2. Stunning chasers or those that try to focus you (mainly carries or anti carries) As you do not have hourglass to save you when your cooldowns are running.

Your ult should be used after at least 2 members are low BUT before the enemy slit up or attempt to Flee.
You should try and be a little late to a fight as not to be the first one focused but not too late that members on your team have started to die.
Your ult is the tool that will pick up the kills for you. while you W and E are your main source of AoE (in a 1v1 like situation single target) damage and you are NOT you teams main form of CC.
Sear is left low as it is the lowest damaging spell in Brands kit in team fights as it is single target.

You are the main pusher. By now your team should have kills and farm so pushing lanes is your job. Clearing minions is a snap for brand as it only takes 1 well placed W to clear a whole wave.
You alone can not destroy tower quickly this is why you push as a TEAM and avoid going solo Unless you are sure you can get away safely if ganked or can easily join a team fight that you where not in to begin with.

Most importantly. Avoiding Dying and be as intimidating to the enemy as possible. this will case them the play defensively unless they think they can kill you. a defensive team play is more reliant on you messing up and handing them the win. DON'T DIE.

Flash is a must for obvious reason it can get you out of most bad fights or get your team kills you otherwise would have lost.

Ignite is important as it allows you to deal the finishing blow to high priority targets E.G. Carries or Anti mages.

however ignite care be swapped for Either:
Exhaust: If you don't plan to take kills from An AD carry or other team mates (Pre-made teams advised only)
Heal: If you want to try and live longer though out the game (and generally be annoying to the enemy)

Reds: Magic Pen is the only choice as there are no other better choices for an AP carry in these slots

Yellows: Mana Regen (flat) for the lasting time in lane.
AP (flat) for faste farm clearing.

Blues: AP (flat) For overall damage and good scaling (advised with Mana regen yellows)
AP (per level) For better scaling in game but less early game damage. (advised with AP (flat) Yellows)

Quints: AP (flat) for great scaling (Opinion: no other better for this slot as an AP carry)

Pros and Cons to this build.

Pros: Great farm
Deadly mid/late game strength
High surrender Rates if pulled off well.

Cons: Hard to kill others early game (this is not for you if you like to kill early game)
More of a Glass cannon as you don't get items like Rod of ages or Hourglass.
Will not and should not be ganking other lanes before lv 8.