Sample Full Build
Nunu is a support tank. That being said, he controls jungle incredible well right now, and can snowball (NICE PUN) to the point of being able to slap around the enemy jungler with Tiamat and Titanic Hydra.
Tiamat is your rush item. It lets you clear quickly, and allows you to actually be a damage threat early and mid game, instead of just a support. That being said, it's not always wise to upgrade to Titanic Hydra. Sometimes, you might want to hold onto it, and sell it later for more tank stats, if you shift to being a support tank in late game fights.
Titanic Hydra. Build it fairly early, or not at all (after Skirmisher's Sabre - Cinderhulk and boots 2 at earliest, and never if you need more tank stats). Only build it late game if you find you're still filling the role of a fighter for your team, and not a support tank.
Ninja Tabi are ideal, although Mercury's Treads are good for countering CC.
Skirmisher's Sabre - Cinderhulk is your jungle item of choice, although Tracker's Knife - Cinderhulk can be considered if you're worried about ward coverage, and not about fighting.
Iceborn Gauntlet is a strong offensive option. It provides much needed Cooldown Reduction, some armor, some mana (which is nice, but not that important), and additional damage and CC.
Frozen Heart is the alternative to Iceborn Gauntlet, when the opposing team is notably auto-attack heavy, and you're less concerned about damage and CC.
Spirit Visage blows your Consume healing through the roof, provides more health, CDR, and magic resist.
Guardian Angel is ideal if you're fighting all-in late game.
Randuin's Omen is a great item for countering crit damage.
Dead Man's Plate is a great armor item earlier in the game, to help you control the map.
Locket of the Iron Solari is acceptable for team-fight utility, especially on a low budget.
Banner of Command fills a similar role if lane pressure is out of hand.
Zz'Rot Portal is another option if lane pressure is out of hand, and you need to help hold gates.
Banshee's Veil isn't as strong as Spirit Visage on Nunu, but does provide a large chunk of magic resist.