Amumu is a champion who depends on his allies, so don't try to solo middle lane. Find a champion who deals a lot of damage at early game because you won't be strong enough to kill some one fast. However you can help an allie kill an enemy with your Bandage Toss.

I think that this is a build which will keep you alive and help you get some kills.

When you don't have any more free space for Guinsoo's Rageblade, sell Heart of Gold to buy it.

Don't buy Guardian Angel if your enemies are noobs and can't kill you.

Buy Elixirs only if you are full build and don't have any free space for other gear.

For me, those spell are the best for Amumu.

Don't choose Clarity if you don't spend lot of mana.

Also Flash is a very helpful spell for saving yourself through jungle and walls but it will also help you chase some low-health enemies or avoid turrents.

You can also choose Ghost or Exhaust for chasing or escaping battle.

Revive and Teleport are also a good choice if you don't want to lose xp because of a death.

Fortify and Heal will help you if you can't defend your lane easily and they push you all the time.

With this champion is hard to jungle because he don't heal and don't have lot of mana but you really want "Big Golem" buff so tell your team to let you take it.

Also, with this build, you can easily kill "Baron" with at least 2 more teammates and tank it, but don't waste your mana and always be ready for a fight.

Amumu is very helpfull for team fights because of his ulti. Use it when 2 or more enemies are near you, if you are with your team mates, to kill them. Avoid to use it for one enemy but do it if it's necessary.