Just a simple AP nidalee build i'm allways playing,and it works pretty well for me ;) Enjoy it :)

I'm choosing runes for ability power and magic penetration,becouse my point of playing nidalee is do tons of damage to fragile targets. I choose seals and glyphs of ability power per level,becouse you get equal amount of ability power on level 6 as with fixed amount of ability power runes.
And also i choose magic pen marks and quints for more damage :)

For masteries i choose 9-0-21,becouse i need mobility,cooldowns,also i get some experience,and addiotional gold, + ability power and magic pen.

I build Nidalee with AP items. This allows her to deal out tons of damage, but i also build items like rylai's crystal sceptre which gives her enough health to not die and/or stay in lane longer. Let's go over my item purchases and why I buy them.

Doran's Ring is a fantastic staring item, giving her a little bit of bonus health which lets her survive. Also it gives AP which is good, and Mana Regen, Which is great on Nidalee because she is mana starved.

Next i get Sorcerer's shoes it is a great item on Nidalee because of the magic penetration allows her to do more damage at the target.

After that i get Rabadon's Deathcap. lots of AP= killing lots and lots and lots of people with your abilities. great item on your heal as well as your damaging abilities because it increases the heal capability, giving you more sustainability and you can escape better with your life.

After this Void staff follows,it gives lots of ability power and magic penetration,so i can do plenty of damage from a max distance,even to those targets who bought magic resist items.I specialy like targets like ashe,who's pretty much easy to damage deadly with this build :)

I choose lich bane,becouse its great of aditional mana,and has a sheen effect,its great using q skill on cougar form.
Ryals scepter is important for me becouse of its aditional health,and slow option,that lets me escape or chase more effective ;)

I prioritize spear at first,then heal,and trap at last,i dont need to do damage with my traps,or bounce in cougar form,i use trap only to see my target if he steps on it,and bounce only to escape or chase.
Spear is most important for me,becouse its a deadly weapon agains everyone ! My heal gives me opportunity to be full everytime,and annoy my targets from the brush with my spear.

My summoner spells are Flash and Ignite
I use Ignite because it is great early game and can get that first kill when they try to run. It can help clean up kills in later game as well when someone has really low health and you need a way to kill them. Lastly it can stop people from healing your target. for example if there is a Annie that is very low health and then you see a Soraka or Sona or any healing champ, cast ignite on the target you want to kill and all the healing done on it is reduced by a very large amount.

I take Flash ,its very good because it allows you to escape quickly, chase people, and also it allows you to flash over terrain. You can get this as a replacement for ghost. This will give you more mobility and you can turret dive then flash out.

You could also use Exhaust. this allows you to chase better and escape better. great on Nidalee. You can get that first kill, or escape from certain death. Additionally, you can use this in a team fight on their carry to stop them from doing nearly any damage.

Pros
Lots of damage from a large distance
Mobility
Heal
Easy to farm after level 6
Trap could be used to protect from junglers
good pusher after lvl 6

Cons
Squishy
No mobility until level 6
Mana goes down pretty easy

Never tried to jungle, but blue buff is allways great idea for nidalee ;)

I suggest to stay in brush as long as possible,you have mobility here,you are not seen,you can still do a damage. In team fights, trow your spear to ad carry,heal your teamate and go into the cougar form to finish them up ;)
Nidalee is a great split pusher. She can easily shred towers with her Primal Surge. The attack speed increase is astounding and when she gets Sheen/Lich Bane, towers fall over even more fast. Now, when DO you backdoor/split push? When you begin pushing, check the mini-map for enemies. Locating enemies are key because if you get caught, the enemy can easily rush a Baron. However, you not being at a fight can force your team to fight in a 4v5 and ultimately possibly lose. Decision making is essential. When scouting the mini map, look at your team first. Are they prepared to do baron/dragon if all of the enemy team magically shows up to kill you? I'm sure you can find great reasoning with decision making but for the most ideal positioning for pushing, you want to find 4-5 enemies in one lane protecting/pushing. Try to recognize your team's positioning. Is it good and difficult to be jumped 4v5, or is it weak and flimsy? If you have confidence in your team and you feel safe, push a tower. If not, push a little bit, apply pressure up top to send a little panic mode through the enemy team's minds. Split pushing is about mind games. If you aren't somewhat experienced at LoL, I don't recommend you to split push hard.

You can use bounce,to jump trough the walls.

Farming is easy after lvl 6,try to jump with bounce into a group of minions,and then use your E skill,to finish them off ;)

Prowl-Prowl is a great innate that allows Nidalee to move faster while in brush. This is very effective for chasing/escaping.

Javelin Toss/Takedown-While in human form, Javelin Toss will do a lot of damage late game, and your spear will do tons of damage.

While in Cougar Form, Takedown is a great ability to do a lot of burst damage.

Bushwhack/Pounce-While in human form, you drop a trap and if someone runs over it, you will gain vision of them, and they will take some Magic Damage. Riot nerfed Bushwhack and it no longer reveals wards :(

While in Cougar form, you will leap several feet forward, allowing you to jump over your small ledges, and if you leap towards an enemy player, they will take some damage. Pounce has quite a short CD, and is easily used for mobility.

Primal Surge/Swipe-While in human form, you will heal someone for an X amount, + a percentage of your Ability Power. Healing someone will also increase their attack speed. This spell is great for helping low allies, and giving attack speed users like Sivir even faster attack speed!

In Cougar form, you will do a small aoe swipe. Great for farming!

Aspect of the Cougar-Nidalee turns into a cougar gaining some buffs, and it allows her to use her second set of spells.

When it comes to laning, Nidalee can play and dominate in whatever lane she wants. She can destroy solo lanes, and she can wreck havoc in duo-lanes.

Solo Lane:

In solo lanes, you have a heal. Heals are vital in solo lane because it'll give you an upper edge against your opponents. Nidalee harrasses very well with her spears and auto attacks, and her heal gives her extra attack speed and health. Using Bushwhack to your advantage, you can help funnel your enemy into an area so you'll have a better chance of landing a spear, or you'll end up forcing your enemy into taking a Bushwhack damage. Nidalee is a very strong solo laner against 1 or 2 people.

Duo Lane:

In duo lanes, Nidalee is also destructive. She can still cover the lane with Bushwhacks, funneling your enemies into a certain area, or forcing them to take Bushwhack damage, and giving you vision. Also, you have the ability to heal, and healing yourself and your team mate makes you even more strong. You are still a strong harrass in duo-lanes, and your spears, bushwhacks, and auto attacks will widdle people's health down. Nidalee is also very efficient against others with a partner. The only downside I see about duo-laning with others and not laning by yourself is that you get Cougar Form slower.

1) Take advantage of Nidalee's innate Prowl! Whenever you run through brush, you gain movement speed. Combine Prowl with spells like Ghost and Flash to make you un-catchable!
2) A great combination to use to catch up to enemies and to finish them off if they're low is Pounce then Flash, then Takedown!
3)Nidalee's Javelin Toss has a long range, and believe it or not, it can snipe people who think they're 'safe' behind their towers.
4)Whenever you are travelling around the map, always stay in Cougar Form, and use Pounce to move a little faster instead of just walking.
5)When you are trying to keep your Guinsoo's Rageblade's stacks, a low-cd pounce, or just switching out of Cougar will help you maintain your stacks.

I will add more tips if I find out more, and if you have one that you'd like me to post, post below in the comments, and I'll give you credit for it!