It all depends on what team u meet, tho, philo stone and hearth of gold is a must, benefits alot over time, with mp5/hp5/gp5. after getting those two items, or one of them, get a pair of boots and mercs or glacial then mercs. after that it all depends on the team u meet. Tho, warmongs is a very nice item due the hp.
Frozen Hearth is a must, gives you additionally mana, cd reduction ,armor and lowers the as of the enemy team, a very usefull item even vs 1 ad.
items really depends on what team u meet, ex: 3ap, you can get abyssal scepter early to have nice damage, lower their mr, and get mr urself. 3handy stats for you and 1for your team.
rylais crystal scepter is also an option vs warmogs armor. hp and hp5 vs more dmg and slow.if your team lack ap, go for the 2 scepters. if not i wud go for warmogs.

for quins you can also go swiftness , 1.5% ms, 3 of them. dodge seals works well if agains heavy ad.
flat mr works aswell tho, mr pr lvl benefits more late game.

Pros:
very nice ganker.
can take alot of damage.
stun.
root.
aoe dmg.

Cons:
easy to harras if lane.
hes small.

as a jungler you depend alot on your team to cordinate a proper or decent gank, even tho they got flash you can still do alot with the right setup. ive found out that veigar and vladimir works very well in combo with amumu, gankwise and teamfightwise, you can easly get fb middle with veigar and amumu, 2stuns, lots of damage. tried alot of diffrent combos but the one working best for me is vladimir,veigar,amumu,taric and caitlyn.
most of what i type here is common sense, you have probably heard it loads of times and boored of it already, ignore what i write, try amumu , test him, hone your skills with him.

depending on your team and how well they do in lane you can go for bandage toss at level 3, to get the stun and a succesfully gank, cordination is the key.

team work is essential for amumu, with cordination you can easly pull of many succesfully ganks, or harras, and quick tower/dragon kills. it all depends on how well you and your team can play with the setup you choose.
for random 5v5 quee with random people its alot harder but you can still do it. talk with your team, cordinate, ping etc etc to get things going, show them that you can play well as a little mummy.

Jungle:
start off with the blue buff(big bad golem) get one to pull it for you or help you get it down alot so your jungling goes quicker, just make sure he gets out of range so u get all the xp, clean out the jungle, ingore lizard. go back and get a regrowth pendant, or see if any ganks can be pulled off, or if any lane needs cover. go back to wolfs, clean again, visit their blue if they aint got a jungler, clean a bit in their jungle. always look for a gank, you can do alot as amumu even if they got full hp, avoid turret diving early game unless very low hp and able to flash out. always keep looking for ganks, cover lanes if needed, dosent matther if you fail to gank, makes them loose xp and gold and possible hp on the turrets.

Lane:
havent really tried it much, tho, last hitting, harrasing as useall, hide and popp out to harras, works very well with a ranged ad.