I liked Amumu so much, I made a guide for him. This should tell you all about how to play Amumu as an effective Jungle tank who is both hard to take down and impossible to ignore.
If you are reading this in champ select, here are some helpful hints no on the cheat sheet (which you should also look at)

• your team needs to help defend against invade
• buy wards
• relax and have fun, you're more likely to hit your bandages
• tell your good for nothing teammates to buy wards

If you want to play Amumu in ranked games, the first step is to find the skin you like most, because an Amumu without a skin is just some useless tryhard who thinks Amumu is an automatic win. I prefer Almost Prom-King Amumu (because I was almost prom king once, and know that feel), but they're all good and this is largely a matter of personal taste. If you do opt for Almost-Prom King, be sure to gift your duo-queue partner Prom-Queen Annie and dance romantically in mid lane while Mr. Tibbers chaperones.

On a more serious note, please don't play Amumu (or any champion, for that matter) in ranked unless you are comfortable with them in normals. Also be aware that the frequency of counter-jungling increases proportionally to your ELO, so increased wariness and communication with your teammates is essential to get through the first five minutes of your games after you crawl out of bronze.

Pros

• wants to be your friend
• incredible cc and initiation
• easy to learn, but with a high enough skill ceiling that you have room to improve
• just press r and cry as your new "friends" in your "group hug" are destroyed by your teammates
• natural resistance to physical damage
• causes your enemies to cower in fear and point fingers at the captain for not banning

Cons

• almost guaranteed to die if caught alone early game
• significantly set back by losing first blue
• yes, your Kayle appreciates the extra mana from second blue, and you tell her it's not a problem and you're happy to help, but dammit you wanted second blue...
• he whines a lot. This takes some getting used to

Amumu's ult may be what makes him banworthy, but his defining characteristic will always be his band-aid. This is your main source of mobility outside of the MS Quints I suggested for runes, and can spell victory or disaster depending on how it's used. For a player new to Amumu, it is primarily a source of frustration, so here are the essentials.

DO

• Turn off smart-cast the first game or two you play Amumu, so you can see the width/range
• Know where the jungle mobs spawn so you can target them without having vision on them (they're stationary, so it's even less difficult than it sounds]
• Target distracted enemies during ganks- band-aids are easily dodged IF you know what's coming. Firing from an unwarded brush gives them less than a second of reaction time, possibly less depending on where their camera is situated
• Target cornered enemies- if an enemy is fleeing at the same speed as you, try to wait until they reach a narrow section before tossing so they have little or no room to dodge without slowing themselves or approaching you/allies

DON'T

• "Spiderman" through the jungle if you need to conserve mana. Faster farming through mobility is great, but running out of tears in the middle of red buff or when there are potentially enemies nearby will get you sent back to fountain in a sarcophagus
• Facecheck over a wall into a 1v5. This requires some map awareness and intuition- did that enemy you're chasing take an odd or slower route to safety to lead you into an ambush? Are 5 champions waiting in that tuft of grass at baron? Nothing is better than hitting a blind band-aid over a wall or into a brush, but if your team isn't backing you up, the feeling of satisfaction wears off pretty fast
• Panic. You'll miss bandages, possibly lots of them. Your team might even start to harass you for it (if they do, mute them), but in 10 seconds you'll have another bandage that's a lot more likely to hit if you aren't still worried about the last one you missed.

Amumu has relatively fast clear times with blue buff; when you hear people talking about Amumu's early game weakness, it refers to his inability to duel with other more aggressive junglers, not an inability to farm. Since the latest patch, Spirit of the Spectral Wraith is the clear option for farm-oriented play styles, as it lets you farm very quickly for extended periods of time without losing mana. You simply don't gain that much from SotAG anymore since it no longer gives tenacity, and its passive is aimed at tanks with slow clear (like Maokai and Rammus), not tanks with fast clears like Amumu. Whichever Spirit item you build, DON'T sit on 80 conservation stacks- this is a waste of gold; conversely, if your conservation is low, you can camp lanes for longer periods of time while building up stacks and "cash in" at golems/wight/wraiths once you are finished, having not lost any gold.

tl;dr if you have (or are about to have) 80 conservation stacks, go clear a camp.

If you buy Spirit of the Elder Lizard while playing Amumu, I will weep for you.

Red versus yellow: Which trinket to buy?
With the new changes to trinkets allowing them to be placed at 30 seconds, you might think Amumu would suffer from more frequent invades, which isn't untrue. The flip-side of this is that you now have the tools to prevent invades. I recommend buying a yellow trinket, warding the buff you won't be starting at (typically whichever buff is closer to top lane; ward your blue if you're blue team and ward your red if you're purple team), recalling immediately for a red trinket, and then going to whichever of your buffs is closest to bot lane. That way, you know if the enemy team starts at your opposite buff, and by the time you're ready to gank, your red trinket will be available for sweeping.


Here are the notes for the situational items listed above:
Randuin's Omen Still an OK item given the useful slow after an ult or a bandage toss, but overall, not as good an item to rush as Sunfire Cape, nor as good a late-game ADC stopper as Warmog's + Thornmail
Locket of the Iron Solari I recommend Locket over Banshee's Veil if you're up against an AP heavy poke comp if your support isn't buying it, your teammates will thank you
Banshee's Veil can effectively be replaced by Locket of the Iron Solari if you're just looking for magic resistance; enemies typically won't waste crowd control on you, but if you're too far behind to tank the entire enemy team, this item can keep you from getting picked off, engaged on, or generally falling farther behind.
Warmog's Armor mandatory if you take the SotAG path (which I advise against), but otherwise an OK option when paired with Spirit Visage or Thornmail
Spirit Visage generally isn't worth it unless combined with Warmog's Armor; Locket does more for you and your team overall in terms of health regeneration, Magic resistance, and cooldown reduction
Frozen Heart should only be bought if you need the attack speed reduction (e.g. versus Vayne, Tryndamere, Twitch, etc.)
Thornmail is a great combo with Warmog's, especially after the Randuin's nerfs. Especially good when Sunfire just isn't enough armor

Since the last patch took the tenacity off of SotAG, Mercury's Treads are the only clear option for boots. If you're up against a team with little CC and REALLY think you don't need the tenacity, you can try:

• Ninja Tabi, for insane tankiness against Auto-attack centric ADCs (who happen to be very popular in the current meta)
• Boots of Mobility, if you want to camp mid and another lane simultaneously to help snowball them (be warned, these boots are quite good early and late game, but come at a pretty high opportunity cost, since you HAVE to buy the alacrity enchantment to get anything close to the sticking power of other boots, as well as losing potential tankiness, magic pen, etc)
• Boots of Swiftness, if you're up against perma-slows (Ashe is quite strong in the 4.11 patch, HINT HINT) and want to have sticking power, as well as some of the overall presence of Boots of Mobility
• Sorcerer's Shoes, if you're snowballing and want to crush your enemies, or if you're the only source of magic damage on your team. Magic pen is a very good stat on Amumu, given his percent health damage on Despair
• Do not get Lucidity boots, as there are plenty of other sources of CDR without the insanely high opportunity cost (as well as up front cost)

But still, Mercury's Treads are your best bet. Tenacity is so important for Amumu for staying on enemies post-initiation and negating disengage, as well as doing damage once in a fight and chasing enemies to clean up.

Amumu is a great champion. Please be his friend, and thanks for being a friend and reading this guide. Even your most inane comments, questions, and suggestions are always appreciated.