I love cake and this is all you need to know. I'm kidding but Amumu is simple. he is a tank.
It might take a while to kill, and you will be getting more assists than kills, but that's a tank's job. The presence of a tank is threatening because they do huge damage over time.

As Amumu, after the fight you want to tantrum and despair but wait before your ulti until you see the enemy starting using their spells on you or an ally. You can use your bandage on a single person if they are all running away. TOWER DIVING AS AMUMU IS SAFE.

Teleport is an essential spell for tanks you should never leave the queue without this.

Flash is useful to run away, initiate or not get KSed. but you won't need this in late game because there is no way you could die, unless if your jumping in 1v5.

Depending on your enemies you might want to switch up your build. For example if your against AD Carries: you may want to getsunfire cape and frozen heart. But at this point you will know what to build on amumu.

AP amumu?
well you can get /rod of ages/Abyssal Scepter/Zhonya's Hourglass/ and then continue on with regular tank builds.

-Bandage Toss[/color] - Amumu tosses a sticky bandage at a target, stunning and damaging the target while he pulls himself to them.
Throws a bandage to target location. If it hits an enemy unit, Amumu will pull himself to the enemy, stun them, and deal 80/140/200/260/320 (+1) magic damage.

Cost
80/90/100/110/120 Mana
Range
1100


-Despair[/color] - Overcome by anguish, nearby enemies lose a percentage of their maximum health each second.
Toggle: Nearby enemies take 8/12/16/20/24 magic damage plus 1.5/1.8/2.1/2.4/2.7 (+0.01)% of their maximum health each second.

Cost
8 Mana per Second
Range
300


-Tantrum[/color] - Permamently reduces the physical damage Amumu would take. Amumu can unleash his rage, dealing damage to surrounding enemies. Each time Amumu is hit, the cooldown on tantrum is reduced by 0.5 seconds.
Passive: Amumu takes 2/4/6/8/10 reduced damage from physical attacks.

Active: Amumu deals 75/100/125/150/175 (+0.5) magic damage to surrounding units. Each time Amumu is hit, the cooldown on Tantrum is reduced by 0.5 seconds.

Cost
50 Mana
Range
400


-Curse of the Sad Mummy - Amumu entangles surrounding enemy units in bandages, damaging them and rendering them unable to attack or move.
Amumu entangles surrounding enemy units, dealing 150/250/350 (+1) magic damage and preventing attacks and movement for 2 seconds.

Cost
100/150/200 Mana
Range
600


-Cursed Touch - Amumu's attacks reduce the target's Magic Resistance for a short duration.

If any of you have questions just ask. ;D GL HF .