DISCLAIMER: This guide was made in January of 2010. Items, champions, and the metagame have changed since then. Take this fact into account.
BT/Stun/Bandage: Bandage Toss
Curse/Ult: Curse of the Sad Mummy
Amumu is a very amazing tank, but is unfortunately very uncommon and tends to be overlooked.
Mercury Treads instead of Ninja Tabi
Situation: The enemy team has massive CC.
Force of Nature instead of Guardian Angel.
Situation: If your enemies suck and can't kill you.
Force of Nature instead of Shurelya's
If you need more MR, sell the Philosopher's Stone for a Force of Nature instead of making it into a Shurelya's.
Amumu's skillshot stun. Important for chasing, support, and can (very rarely) be used as an over-the-wall get-away. Learn to aim this well.
Amumu's anti-HPstacker skill. This is amazing against everyone, but even more so against high-HP enemies. It is also very useful for Lizard, Golem, Baron, and Dragon.
Amumu's main skill. Passively makes him tougher, and actively great for farming and killing. Learning to hurt multiple minions evenly so you can get many kills at once with Tantrum is very helpful.
Curse of the Sad Mummy
The icing on the cake. Curse is probably the most valuable skill in LoL for team fights when used properly. If you're not initiating, which you should be, you can easily just Flash into the middle of all the enemy champions and pop Curse to snare all of them.
Conservative farming is what Amumu is all about at the beginning. Don't be afraid to use your Tantrum just to kill-shot 1 minion. You need all the cash you can manage.
Still should be farming, but you also need to be helping teammates out with being a gank initiator and/or lane helper.
You should be a godly tank by now with your cape, Veil, and GA/Force of Nature. You also should be destroying people while they (if they're good) focus on your carry(ies) and/or casters: stunning, Tantrum spamming, sunfire degen, Despair degen, Curse if there's 2 or more of them.
A very underused and overlooked champion, Amumu is an amazing tank.