This is my first build for any character so I decided to choose one that I normally do good as

First of Amumu is very dependent on teammates in a tank build he does some damage but not a tremendous amount so amumu should not solo find yourself a good teammate and you should be all good. 2nd you better be good with bandage toss realize that if the bandage goes through a minion you will be pulled to it instead of your intended target. Also use despair before entering team fights or pulling yourself to an enemy so it will do more damage. Lead your enemies with bandage toss I cannot tell you how many times I've seen amummu's miss when they try to throw their bandage at an enemy that is fleeing because they don't lead their target

Also on a side note Amumu can be effective as an ap character can do a lot of damage with his ap ratios: bandage toss has 100% ap ratio, despair is .005% ap ratio which seems very low but it can be very damaging at high amounts of ap due to amumu's two stuns, tantrum is a 50% ap ratio, and curse of the mummy has a 100% ap ratio
according to http://www.leagueoflegends.com/board/showthread.php?t=15812

Pro's
Insane hp and regen
Two stuns
Can change the game

Con's In tank build
Dependent on skill shots(bandage toss)
In my opinion pretty expensive build
Dependent on teammates
Teammates tend to get mad if enemy dies to despair

Get Bandage Toss at level one so that you have the opportunity to stun and grab an enemy then if you believe that you and your lane partner can kill them then you can use exhaust as they try to run away for an early game kill or assist.... but don't be too greedy, realize that the grab will only do 80 damage plus 100% of your ap which will not be much at level one. This is also a good way to cause a lesser skilled player to use one of their summoner spells (i.e. flash).

Tantrum at level two and three to help last hitting minions, due to low basic attack damage and speed you may find it hard to last hit minions without this in early game. this can be cast right after an attack so use that two your advantage

Bandage toss or despair at 4 depending on if you are getting attacked by several opponents this early in the game. Despair if your getting ganked or bandage toss if the enemy is leaving you alone in your lane.

Get the opposite of what you got at 4 for level 5 preparing to get your ult skill

Level 6 is pretty standard get your ult. You have a couple of choices on how to use it.
a) opponent is in ult range already and your lane partner/teammates whatever are ready to go. so use despair and then your ult to stun, tantrum and beat the hell out of your enemy and then bandage toss as they try to escape and beat them up some more if they are still alive.
b) opponent is a little farther away. This has the same principle as part a just in a little different order. Despair first, bandage toss to them to stun, tantrum and then ult as they try to run.... if u have a lane partner that can also stun, knock in the air, etc. it may be possible to bandage toss them again if necessary
c) If you have the teleport summoner spell and an enemy is pushing forward in their lane. First inform your teammates that you plan to teleport behind the enemy (specify if more than two) and stun them so that when you do you don't get left on an island with an opportunity for death. 2nd teleport in and turn on despair then bandage toss to the enemy that you specify. 3rd use tantrum for more damage. 4th ult to stun as they try to flee. If your teammates that you are trying to support have a stun or other skills that can trap a unit, bandage toss again when its off cooldown and if your opponent is still alive.

Other than that skill build is pretty up to the player just level up your ult whenever possible.

Ghost so you can get to a fleeing enemy and bandage toss to kill
Exhaust to prevent fleeing
Teleport explained in the skill sequence section
Flash to get in skill range or out of trouble
also Ignite I guess could be useful in order to finish an enemy that is fleeing

Nothing else could be useful. Period that's it no exception

Begin with Regrowth Pendent
on 1st comeback grab shoes and upgrade pendant to philosopher's stone
2nd comeback finish shoes and grab 2nd regrowth pendant
3rd grab the giant's belt
4th finish warmog's
5th whatever part of sunfire cape
6th grab other part of sunfire
7th finish sunfire
8th archangels for mana and regen
.
.
.
Finish Build
If you don't like the reverie you could sell the philosopher's stone to get banshee's veil

You should engage in team fights due to high amounts hp, hp regen, and relative armor
Use your ult to stop enemies in their tracks and give your teammates a tremendous advantage
ALWAYS BE CRYING
Tantrum whenever possible because cooldown decreases when amumu gets hit
Bandage toss enemies when possible.
stop the fleeing enemeies with bandage toss for maximum enemy casualties.

Amumu can be very fun if played right. A strange satisfaction is obtained crying on your enemies and them not being able to kill you. This is my first build so whatever sort of criticism you wanna leave is fine with my and HAVE FUN AS YOUR EMO MUMMY :)