Generally i play a lane amumu, i'm not a big fan of jungling him, but he does make a good jungle, usually i'll pick up BT first for the extra little harass and the stun early game is really nice, and from there i will max tantrum, ult, despair. The damage of tantrum is great, plus the fact that the cool down is lessened by being attacked, and it makes you take less damage really makes it far better than despair.
As for build i generally go with philo first(getting the health regen portion and a health pot to start) however i never build it into anything, i keep it for the extra regen and gold ramp and at the end of my build will end up selling it for a better item. after this i build the boots that i feel will best suit the game usually with swiftness or mana resist(not a big fan of ninja). From there i will usually rush SFC, not only is it a great item, its even better on amumu, and really help ramp up his farming and damage output. From there i usually get abyssal scepter, its nice for the magic resist plus it gives you a great amount of utility as well as damage output, making it so that your actually a threat and the other team has to focus you(no sense being a tank you dont have any damage output and the opposite team just ignores you). After these initial 4 items which i almost always buy, in this same order regardless of the game, my build will change depending on the opposite team, i usually never end up getting warmogs, but if the other team is more magic based i'll get a FoN and possibly even a Banshees veil. If the team is more dps heavy i'll usually pick up a thornmail and rylais. There are also times where i will go the less tanky route and get a rylais, death cap, and some CD reduction, and be a far more heavy hitting death machine.
Hope this helps at all, its the first guide of any sort i'v ever done for a character, and i know its not perfect, but i do quite well with it, and it is the reason why amumu is my favorite tank to play, he is very versatile, and can actually be threatening enough where the other team has to focus him, and his BT+ult or flash+ult is just an AMAZING way to initiate any team fight.