I start out with a Doran's shield and hp pot and go jungling. YMMV.
You WILL need:
Hp regen: I use a regrowth pendant.
Boots: Start with basic ones.
Mana regen: start with a meki pendant.
Thats your basic items. Dorans regrowth boots meki. Let's start upgrading.
First, look at the opposing team. -What are their damage sources and how many non-knockup, non-silence disables are they packing?-
80%+ of games I go mercury treads, 20% ninja tabi. Only two boots worth it on amumu. If the enemy is almost exclusively physical DPS and/or is very disable light, go ninja. Otherwise, mercury.
Now you need to grab your chalice. Next up is aegis of the legion (perhaps you started building this earlier dependent on the game). You're starting to become a respectable tank. Note: if you are doing very poorly and are very underfarmed, just start stacking hearts of gold, perhaps even eschewing earlier items. Hearts of gold are your fallback item if you're just getting raped. Hopefully you can last til lategame with a full set of hearts of gold and start turning them into real items.
Now, assuming you're doing well, it's time to get serious. You should have a doran's, regrowth, mercury, chalice, and aegis. Here are possible items:
Guardians angel: I get this in 70% of games next. Armor and MR are far more useful than HP, plus its passive is awesome. Against a balanced team, grab this.
Let's say the opponents are flabbergastingly dumb and are almost exclusively spell damage. You have a choice between banshee's veil and force of nature. Banshees veil: hp and spellblock. FoN: slightly cheaper, slightly more MR, and hp regen.
More often you will want to get a physical item after your GA. good choices: sunfire cape (by this point the passive isnt that great, but take what u can get). Wardens mail.
About the only item I don't really ever get is warmogs on amumu. Others may swear by it, I prefer having 2500~ hp and RIDICULOUS armor and MR (about 300 of each).
This is pure tank amumu. The reason amumu is so awesome is he is dangerous with just a tank build. His q w and r just demolish enemy teams whether they focus him or not. Even without AP I find enemies have to target me or they just end up getting wrecked. On the occasions that I am -too- tanky (they literally just can't touch me and give up and go for my teammates), I do grab a rod of ages/sunfire early. This is rare.