Pros:

Good Initiate
Great Survivability
Passive is good against low MR opponents or if your team has good amount of AP
Ganks can be godly if one can land bandage toss

Cons:

Has to be right in the fight, if not durable you are an easy kill.
Melee oriented
His Q, what makes him so good, is a skillshot

For Amumu you always start Wolves/Blue, no exceptions. With your W you can quickly clear Wolves camp with help from your Top/Bot and Mid lanes. With enough damage, Blue buff can go down in no time as well. You will then head to Wraiths and either back to Wolves if they have respawned or to Golems and then Red.

At either 3 or 4 you will pick up Bandage Toss and gank a lane. Gank only the lanes where the enemy is low or has pushed your lane towards your tower. Be sure to ask your lane if they have warded and where so you can avoid being seen.

If you push mid or bot far enough back, tell your lanes to go for Dragon.

Amumu's Q is what makes him great. It is a skill shot, however, so you will need to practice landing it. Some cool tricks to try: if you are being chased you can bandage toss to where you know a jungle camp is beyond a wall to escape. You can do the same for minions in lane. When trying to land your Q go for a high damage target like their ADC or APC or one who can shut down an initiate like Leona or Sona.

Amumu's W is toggable and especially early game you need to be aware of when it is on and to turn it off before you run out of mana. Try and have W active before initiating a fight.

Amumu's E is great for the jungle as it reduces physical damage slightly and its cooldown is reduced for each basic attack, making it almost spammable when jungling. It also adds that little bit of beefyness for teamfights. When initiating a fight, be sure Tantrum is available to use.

Amumu's Ult is what makes him banned and scary. A large AOE stun that does plenty of damage as well. Use this right after your initiate when most or all of the enemy team is within its range. You can also use this as an escape or saving a teammate. It has a fairly long cooldown, so try and use it only when you know it will be used effectively.

The reason Amumu's core items always consist of Abyssal Scepter, Sunfire Cape, and Rylai's Crystal Scepter is because of the huge damage increase it gives to Amumu. Abyssal Scepter's range is a little larger than Amumu's ult, giving all of his abilities the affect of reduced MR. The Sunfire Cape adds to Amumu's high AOE damage and works well with Abyssal Scepter as Sunfire Cape does Magic Damage. Rylai's Crystal Scepter gives a great amount of AP and slows enemies who are affected by Magic Damage. Also, all three of these items give Amumu good durability with MR, Armor, and Health.

When jungling, you upgrade your Machete to a Spirit Stone and upgrade that to either Spirit of the Ancient Golem or Spirit of the Spectral Wraith. You will go for Golem if you are the tank or if you will not be getting Mercury Treads and Wraith if you will be doing more damage. Late game you should sell these items for other recommended items, either being centered on more damage or more tankiness depending upon the situation.

Any other items you buy are based purely off the situation. If you need more defenses you need to figure out what you need.
Do you not have enough Armor?

Randuin's Omen
Zhoyna's Hourglass
Frozen Heart
Guardian Angel

Not enough Magic Resist?

Runic Bulwark
Twin Shadows
Banshee's Veil
Athene's Unholy Grail

Do you have enough resistances, you just need more Health?

Warmog's Armor
Rod of Ages
Runic Bulwark
Randuin's Omen

If you are in need of more damage because you have enough defenses or you are not the primary tank you should consider

Void Staff
Rabadon's Deathcap
Will of the Ancients
Liandry's Torment

With these offensive items need to decide which is better for your situation. You should get Void Staff if the enemy is building a lot of MR, get Deathcap if not. Will of the Ancients gives you some spell vamp, which isn't that important for Amumu, but if you have allies who could benefit from the aura, then purchase this. Liandry's allows for more damage over time to be applied to those within Amumu's reach, it also gives a nice amount of health and spell penetration, so purchase this for a bit more tankiness and against enemies with some MR.


Remember that some defensive options still provide you with good amounts of damage. Zhoyna's Hourglass, Twin Shadows, Athene's Unholy Grail, and Rod of Ages all provide defenses, but also give AP.