Other posts cover items, builds, masteries. I thought I would talk about some of the basic tactics and tips for playing Nunu, mostly I play AP Nunu so you'll find tactics that reflect that.
Also I'm sure pros will know all this stuff this is more for people new to Nunu (or maybe people who fight him who want some tips on what he should be trying to do to them)
Play Aggressive with champions
Nunu has to be played aggressively. Nunu's passive power allows him free casts after hitting creeps. As such you need to be hitting creeps. Which means your opponent needs to be pushed back. Ice blast is great for this. Get this level 1 and start to harrass immediately. Get a health pot or two also to tide you over until you get consume. Nunu does take damage if you play him agressively.
Play kind with creeps
AP Nunu is quite poor at killing creeps. He's got no AE (You're not going to use your ulti on creeps) and while his consume is good at eating the boss minions its still not fast pushing a line. So don't push the line. Hold the the line in your own half, build up your passive from hitting creeps then relax. Just try and last hit mobs.
The reason you want to do this is because Nunu is a menace and very often you can push the enemy back to the point where they can't hit the mobs to get last hits. If you're in your own half they won't get exp either as they'll be at their turret. Also its safer for you if a gank comes in.
Nunu is great at running people down with snaring ice blasts. But you need distance to do this. This is very dangerous if you're fighting around their turret, but easy if you're in your own half.
Rely on your partner and turrets
Nunu will harrass the enemy well given half a chance. But if you solo against a Teemo or various other ranged classes you're likely to get shut down. If you partner Nunu then your job is to help him stay in the creeps. Apply pressure.
If you are getting pushed and they can stop you consuming you're in trouble, just fall back to the turret, take it easy. Consume there and wait until level 6. Then Ultimate them. Your first use of ultimate is very often a free kill as people don't realise you have it yet (they watch their levels not their enemies). You also have a long way to run them down from your turret, also good for you.
Ultimates don't draw agro
When you channel your ultimate you don't draw agro. This is important in two situations:
1: Using ultimates under enemy turrets: Everyone has been there. The enemy is at 30% health but hes sitting under his turret, while you watch your creep waves getting wasted there. At level 6 you can stroll in and ultimate. This surprises a lot of players and they die. You can run out at that point getting only a minor lashing from the turret after the fact. Obviously if you ice blast on the way in you'll draw agro. So your first damage on their champion has to be the ultimate
2 Ultimating in bushes: You can't be seen in bushes. At the team ganking stage when fights happen in the jungle you will find you can ultimate in bushes and people won't know you're there to run or to interrupt you. This is one of the most decisive ways to win a team battle. Teams with Nunu should really consider luring into jungles and fight off the lanes for this reason alone. Again, don't give the game away by Ice blasting first.
Use Boiling Blood on the Melee Carry
Okay really obvious. but Boiling blood gives movement but also up to +60% attack speed. Thats massive for a melee. Many Yetis just Boiling blood themselves. Don't be that Yeti. This is one of the best buffs in game. Your carry will be much more effective with it.
Also if you're teamed with a melee at the start drop one level of Ice Blast for boiling blood around level 3-4. Keep your melee buffed. There's a lot of Nunu builds which suggest getting the first level of boiling blood after everything else, This ability is highly highly underrated, itself effect on your melee team mates is large.
This is a must summoner ability for Yetis. Most abilities are either useful start or end of game, flash helps Nunu in both. Early on its great from escaping turrets after you've ganked someone (or just escaping ganks). End of game its great for flashing in and setting up ultimates quickly.
Don't Jungle early on.
This is opinion, but theres quite a few guides suggesting Nunu should jungle. mainly based on him having consume and his passive. Don't bother. The reason for this is because Nunu is a great early game killer. When you have a good partner you've very likely to get early kills with Nunu. Running people down using Ice Blast is very effective. You'd do better getting some early kills than maybe slightly increased exp. Also items like Mandreds aren't as helpful for Nunu in the team game stage.
Own the Jungle late on
Nunu is devastating in the Jungle. He can get close to people easily and use bushes to channel in safety. Any team that doesn't feel safe in the jungle is at a disadvantage. They'll probably play defensive at that point and its hard for a defensive team to win. If you own the jungle you can pick off creep groups and turrets. If you fight up and down lanes with Nunu you're not making best use of the terrain
Remember you can stop ultimate early
Your ultimate can be ended at any time to do a percentage of the damage based on the catsing time. It seems many Nunus don't know this. Its common to see Nunus channel as their opponent walks out of the blast area. Just end it early, obviously some practice is needed to know its max limit to cancel it as late as possible.
Learn to kite
AP Nunu's have poor melee damage output, often you don't want to go 1v1 with someone when you could be kiting (obviously theres also times when you want to tank someone's last block of health). Good examples are melee with life steal. You'd do better kiting these champions than ice blasting them and letting them get health back off you while you wait through the cd. Someone like Yi you're better kiting as long as his ulti isn't on.
For AP Nunus you should consider getting these items
Mejei's Soulstealer - This is great for Nunu because you're hard to gank, so you won't die that much and your ultimate willgive you a lot of assists in team fights meaning you'll power it up quickly. Often I find I've maxed this item. Thats extremely cheap AP. If you play Nunu like a suicide bomber though this item probably isn't for you.
Banshee's Veil - This stops one hostile incoming spell. This means you're less likely to get CCed when you're channeling your ultimate. This is useful against good enemies. Personally I prefer just to be more careful about when I use it though than getting this.
Ultimates aren't just about damage
In a team fight if Nunus use of an ultimate means enemies start to run then they aren't attacking. Also they're heavily snared. Any team should exploit an ultimate use quickly. Psychologically it turns the tide of a battle. Make sure, as a team member to Nunu, you capitalise and lay in whilst the ultimate is being channelled.
Look out for your team mates
Nunu is great at helping people escape. Boiling blood, plus an ice blast on a choice enemy, will often save a team mates life. Everyone loves a guardian yeti-angel.
A Yeti in the bushes is worth 2 in the open
Nunu works best close up. Nunu is one of the most effective classes for fighting in and around bushes. Hes much easier to counter when you know he's coming. You will find yourself getting CCed much less when you start from bushes and jump out in fights than when you mill around in the open waiting for a team standoff to resolve and everyone's hovering over their cc buttons and looking your way.