Who Am I?
Well, I'll be blunt. I'm just a Silver player that plays quite a bit of Amumu. I started getting S grades with the little before I was level 2 with him, and earning Mastery Level 7 with him took a mere 2 days because Riot handed me more S's for Amumu than I have total for all my other champs combined. I'm no authority in League, but my Amumu build WILL win you games.

Why Amumu?
Amumu is a very solid low ELO Jungler. He is simple to use, very fun to play, and very powerful in teamfights. He is very easily countered, but to Bronzies that don't know what to do, Amumu will dominate the Rift.

Smite is essential to all Junglers. Without it, you level slower and have less sustain. No further explanation should be needed.
Flash is a mainstay to most champions, and our mourning mummy is no exception. Initiate, escape, set up dragon/baron thefts, Flash does it all! With style!
Ghost is another solid pick on Amumu. Being able to keep up with your opponent while your Despair eats away their health is a very important aspect of Amumu, but the lost versatility by not taking Flash is a looming factor of picking Ghost.

With no suitable replacement for Deathfire Touch, Amumu is forced into the support tank role. This isn't that bad of a thing, as it is a role Amumu and the new runes thrive in (RIP Acid Tear Amumu, though).
Aftershock is and obvious choice for Amumu- It benefits champions with hard CC and those who want tankiness, both of which are traits that Amumu has. Our goal as the mummy is to support our team while our enemies slip on our tears. Therefore, Font of Life gives your friends a little extra health for attacking the champions you lock down.
The third tier of the Resolve tree is a toss up. I prefer Iron Skin, as Amumu's early game is when he is weakest, and the armor will help with your clear. Conditioning is usually better, as it gives better stats, albeit later than the other two options. If you are confident in your early game, by all means, take Conditioning.
You don't heal or shield, so obviously you should take Revitalize. If you want to win, though, taking Second Wind will give you psuedo-tankiness when you gank or 1v1 the enemy jungler.

As for you secondary tree, I prefer Dominations Cheap Shot and Zombie Ward. You have movement impairment for days, and thus true damage for days. Zombie Ward lets the wards you leave around the map have double duration, something you laners should appreciate.

Taking Sorcery as your second path gives you access to The Ultimate Hat and Celerity. As I will detail in the next chapter, Amumu's ganks are reliant on his R, so The Ultimate Hat feels really good on him and allows for more ganks. Celerity feels great on any jungler, as moving faster means clearing faster which means more cs, more gold, more items, and more efficient ganks. Any questions?

I personally do not feel like Amumu is a reliable ganker without Curse of the Sad Mummy. You may disagree once you get used to him, but to each their own. I do not often gank without my R up, and pinging its cooldown is how I signal ganks. Once you hit level 6, gank whichever lane you feel needs the most help, using your R and Q to keep the enemy still long enough that your W and E spam should kill them without your laner helping. [Note that your laner should help.] Actually, try to let your laner get the kill. This will decrease their dislike of you for afk farming. Once the gank is over, go back to farming for 150 seconds (or less if you take The Ultimate Hat) unless it is absolutely a necessity to help other lanes. What do you do in this time? You can solo dragon at level 6, and solo Rift Herald at level 9 (provided you have blue buff). Grab these objectives in between ganks, and your teammates should not be very upset that you've only ganked a handful of times.

So you've been afk farming and are approaching full build 28 minutes in, have several assists and every dragon under your belt. Rift Herald killed and used, and everyone is itching for teamfights. What is your role in them?
Simple.
Make the enemy team cry harder than you do.
Bandage Toss onto a choice enemy (usually the ADC or fed midlaner). Ult. Smack as many different people as you can to apply Cursed Touch and deal constant damage with your Despair and spam Tantrum. Your teammates will hopefully follow you in and help you clean up. The constant percent health damage with W combined with your ability to stay alive in the center of the fray will lead to an easy teamfight for you, provided your team backs you up. Combining your kit with other AoE ults (such as Illaoi's Leap of Faith or Zyra's Stranglethorns) leads to devastating teamfights.
Ez game, ez life.
If it hasn't been reinforced enough, Amumu is very reliant on his team to kill people. Runes Reforged stripped Amumu of the damage I used to take into the Rift, and he is forced to play into the support tank mindset now. His teamfight is still very strong because of his ability to affect the entire enemy team, but the damage is significantly less than what it once was.