Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Step 1: Start off with Dorian's Shield + Health Potion. Start off at mana golem. To kill mana golem you can smite, auto attack until you take two hits from the mana golem, then pop consume and autoattack the mana golem until he dies. If you will need the health potion (due to lack of runes), make sure to pop it right one second after you consume. With these cooldown reductions you should be able to consume a small mana golem camp minion right after you kill the mana golem. You can attack the other minion to death. This should tide you over to lvl 2. You then move to the wolf camp and take that. A consume + one attack will kill the main wolf. You should be able to consume a small wolf after you finish autoattacking one to death (with maybe one cast of ice blast). You can then take banshee camp. Consume the main banshee and autoattack/iceblast the other minions. You should be able to consume another minion. Smite will be off cooldown by the time you then move to the lizard camp. You will probably have about 2/3rd of your health as you approach lizard camp. This is normal. Smite, iceblast, consume as needed. You should only need to consume the lizard once, and spamming iceblast/autoattack will take care of the rest. You can then consume a small minion at the camp to heal.
Step 2: There is where the interesting choice comes in. You can A) go for the 2 golem camp or B) go in the other team's jungle. You will be lvl 3 at this point. I usually prefer to go to the other team's jungle. If the other team has a jungler, you could go check their blue or red. However otherwise i would go for their wolf camp and then their mana golem camp. Then you have to walk the long way around, taking your banshee and wolf camp. You should then be lvl 5 and ready to take dragon. This should be around the 5-6 minute mark ideally. You should have almost full health after you consume the dragon at first. Spam bloodboil as necessary, use iceblasts, and consume. Note: You will reach dangerously low health when dragon hits 1k hp. This is where the CD runes matter, especially if you don't have all runes. You should be able to consume the dragon to kill it and actually needing the full true damage value of consume. You don't need full runes to accomplish this. 2/3rds of these runes will work and 1/0/21 masteries will work (with 1 point in smite). At this point you will have around 1.6k. Some lane fights will definitely be going on and you can go appropriately to help. Ultimately the goal is to be able to buy rod of ages before you have to ever recall back. Now is where a perfect description of all your actions end. You have to judge what to do essentially, with the exception of taking dragon when it reappears if possible.
Rationale: Although Nunu's passive preserves mana and we have 1 mana/s from masteries, the golem buff allows for cooldown reduction as well and is overall the best choice to start with. You should have 100+ HP if you popped the health potion even with minimal to no runes. Dorian's Shield is taken as a first item because of its versatility and usefulness. This is a very anti-spell focused build. Mercury threads and Banshee's provide MR, as such even a little armor helps. Doing mana golem 1st is a lot harder with dorian's ring and will probably require full runes and masteries. We can skimp on buying boots right after Dorian's shield because bloodboil provides a run speed boost, flash allows you to escape, and iceblast can slow any single champion trying to chase you. We buy rod of ages so it will obtain its full effect faster. At any moment after you get rod of ages, you may decide to get elixirs of brilliance for additional CD reduction so you can spam iceblasts to do some damage in team fights. Often when buildling for banshee's you may choose to buy elixirs in between. These elixirs are to provide you with a temporary advantage to make up for the lack of extreme AP in this build. Mercury Threads may be substituted for Sorcerer's boots but I believe the extra 25 MR > 20 Magic penetration especially given the lack armor/MR in this build. After banshee's veil, i would usually buy elixirs of both fortitude and brilliance and build an optional item that may depend on the situation. Although Nunu will have over 4k HP, his armor will probably be lacking. As such you will probably want an armor item or another HP item. As for summoner spells, smite is taken for obvious reasons to optimize jungling. Flash is taken to escape from dragon if necessary and to escape a gank in the jungle and overall in the game. Flash is especially important because we avoid buying boots for a long time and since blood boil doesn't equate to the speed of boots.
Actual Gameplay that isn't jungling : there isn't much here except to say to use ice blast on the appropriate targets in team fights, timing your ultimate with a teammate's stun, slow, fear of some sort. You can disrupt your ulti to do minor damage if they are about to escape. Bloodboil your teammates to help them do a variety of things including escaping and giving your physical DPS more attack speed. This works great especially with someone who buys items with on-hit effects like Madred's or champions with per hit effects like Kog' Maw.