Early-game:

Often I find myself in the scenario where someone has been harassing for a while early game and I decide to NOPE them harder than that one time they really wanted the cool action figure with Kung-Fu grip. I would Bandage Toss, cry my life off, throw a Tantrum, curse them, and if I timed it correctly I should be able to stun them again before they are able to run away leaving them with another Tantrum and a few more seconds of Despair. Depending on my laning partner they will be either super dead, really dead, or at 25 health. If they are extremely low health and you are still almost full you can go in behind their turret, Bandage Toss (don't miss) and either 1-hit them, or Tantrum and flash out of turret range.
Although this seems extremely unlikely it's really easy to pull off against morons that overextend, most of the time it's a good thing to let them push you to turret, just don't let them do significant damage to it. *Level 6 minimum*

Also, when you are facing heavy melee auto-attackers that are overextending, I.e. Master Yi, Tryndamere, Jax, you are able to Bandage Toss and then Tantrum and run back to safety without being damaged. This is a good way to harass with Amumu at a low level without taking much damage. At this time you should have already ported out and back in for your Philosopher’s Stone and should have the Health Regen and Mana Regen to heal if you got damaged a bit in the process or get damaged when using Tantrum to farm.

The entire goal for the early game as Amumu is to farm as much gold as possible and try to help your laning partner get as many kills as possible.

Mid-game:

During the Mid-game you want to focus on getting the Golem Buff as much as possible, ganking as many people as possible, and pushing as many turrets as possible. Team fight are a must in the Mid-game and you are the key to winning those. If you can time your Curse of the Sad Mummy properly your team should be able to take out at least 3 enemies in a team-fight. If things go bad you are pretty good at escaping due to your stun, ability to pull through walls, and do heavy damage to anyone dumb enough to follow you extremely closely.

End-game:

Delicious End-game, as Amumu you are pretty much the main target now, but if you are far enough into your build you should be strong enough to take a massive chunk out of turrets and enemies. Keep on pushing and keep on being a key part of the team fights is the key to victory. Amumu should be able to 1v1 any champion on the enemy team and by end-game can duo Baron Nashor with any other champion in the game. You are Amumu, The Tank Slayer, wear the name proud and don't let anyone on the enemy team push you around.

Pros:
Amazing initiator
Amazing farming
Can tank and have good DPS at the same time

Cons:
Squishy Early-game
People will often tell you that you are bad early game
Many kill-steals, accidental or not.
Banned in ranked, most of the time.