This is my first guide and i dont see any other builds, that build Amumu like i do.
This build is about the build than about learning to play. There are other great guides on how to play him and i reccomend reading those aswell.

Please leave comments on how i can improve this. I will try out your suggestions and add them if i think they are helpful.

While i lane with Amumu i chose Exhaust and Teleport.

Here are some other summoner spells that you could go.

Flash
you can Flash over minions to Bandage Toss your enemies, chase or run over walls. it can set up your ultimate easier. i dont think it is neeeded but alot of people like Amumu to have this.

Cleanse.
This is anti CC. you are more effective while you are able to keep between your carry and there team.

Ghost.
Good for chasing and fleeing. Ignoring minion collision is pretty nifty and can help you alot.

Teleport.
i like Teleport for the reason i can get straight back into the game when i get my items. i can teleport to our turrets that are being pushed. teleport to your Wards for surprise gank.

Exhaust.
you almost waste a point in the offense tree if you dont grab a point in Exhaust or Smite. but as i prefer laning i tend to go for Exhaust

Smite.
this is a spell you take if you are going to jungle, you could also take heal instead of this. but it is not as quick but a little bit safer. Still good spell though :)

Heal.
You are able to jungle with this instead of using Smite. i believe this is a bit more useful late game in team fights compared to Smite.
While it is slower, you are less likely to be ganked for jungling. you are able to take the mastery for this in defense to get the cooldown lower. If you chose to lane, this can also help for First Blood. Most people forget about the heal and go all out if they think they got the upper hand.

Fortify.
I will only take this in 3v3. while it is a good spell to have in 5v5. i just dont think it is as good as Teleport or Exhaust for Summoners Rift.
While Teleport is not as useful in Twisted Treeline i grab this :p

Marks are a no brainer.
Magic penertration is the best choice for spell casters and Amumu is the same.

Seal are about personal preferences
I go flat armour. You could also go with health regen, mana regen or magic resist.
i chose armour for the sole reason for the laning phase. i start with 40 armour which is quite nice and helps keep minions dmg down while hassling.

Glyphs
I take 8 CRD per lvl Glyphs and 1 magic resist and i have good reason for it
to max out your CDR. you only need to have Frozen Heart and 8 CDR per lvl Glyphs (puts CRD at 30.66%0, then when you buy a Blue Elixir that pulls it to the max at 40%!
Having a last CDR Glyph would be pointless, so i put in a magic resist Glyph.

Quintessences
I like to have extra run speed.
Health helps at the start of the game and doesnt stay as helpful after you start lvling. but the run speed will help during late game aswell.

Starting items9
I like to start with a Regrowth Pendant and a health potion.
I lane until i got enough to complete a Philosopher's Stone

Boots
Most of the time i go Merc Boots
Only if they are heavy AD, i will change to Ninja Tabi (Dodging still helps your CD of Tantrum, so dodge is still viable!! XD )
You might want to grab Boots of Swiftness if you dont want to have movement quints.

Armour
most games you will have enough armour by getting a Sunfire Cape and Frozen heart.
The Sunfire is the Next in my building list because it helps with laning because of the damage.
I have put extra armour items in the build that i would take if i am dying to easily in team fights.

Magic resist.
My main magic resist items would be Force of Nature and Banshee's Veil
other good items to take if no one else has are Abyssal Sceptre and Aegis of legion
In some cases where they have alot of cc, i would buy a Quicksilver Sash. it has a active which removes all debuffs from you which can turn the tide against some teams.

Damage.
to be a successful tank. you still need to be able to put out enough damage that they are forced to attack you first, and you can survive the enslaught while the rest of your team is destroying them.
For this i choose a Rylai's Sceptre. it gives you 500 health for extra tanking, a sweet permanent slow with despair (only 15%, but that is extra CC to help keep your team running and setting up against them). and 80 AP, combined with your Blue Elixer that puts you over 100 Ap which gives you an extra 1% on your despair. making you alot stronger over time and a bigger threat. Great item for Amumu! :)
If their team is AP heavy. abyssal Sceptre is also viable. but only if you dont have an AP carry doesnt want to grab it. So ask first then grab it :)
Void Staff is a good items if they have got alot of tanky characters. with Void staff's Magic Penetration and runes and masteries. you can get there magic resist down alot.

Say if they have 200 mr.
your passive reduces their mr by 35 to make it 165.
your runes take out 8.55 reducing it to 156.45.
then your Void Staff and masteries make you have 55% mr pen!
taking off that 55%, you leave them with 70.4 mr!!

Thinking that they have 66% dmg reduction, with your penetration they only have 42% dmg reduction!
that means you will be doing an extra 50% damage of what your meant to be doing! that is alot of extra damage.

You are able to team up with anybody fairly well.
Here are some partners that are outstanding.

Fiddlestick.
Fiddles passive also reduces MR, at end game you and fiddle reduce there MR by 55 with only the passives.

Soraka.
Soraka's heal can make you play alot more aggressive, you do alot of burst damage and most people will run once you done your burst while hassling. so you can keep them off lane doing this a few times.

Kayle
Same Reason as Soraka. But she has a slow and can do a bit more damage.

Poppy.
You can stun someone to give Poppy time to set up her charge. giving you better control over the hassling.

Alistar.
Same reason as Poppy, can push them under the turret for a long time. and if played right you can have your Banage Toss ready again by the time your combo has finished. early game, this will garauntee you a kill.

I cant think of anyone else that will be good.
People who deal magic damage are well suited.
Skill shot characters benefit from your Bandage Toss,