Since the nerf to Primal Surge, I now opt to max Javelin Toss / Takedown first.  Since Nidalee's Cougar Form abilities do NOT grow in strength until Aspect of the Cougar is leveled up, you will only benefit in your human form abilities when leveling them.  

Javelin toss damage is calculated by how far you are from your target when it makes contact.  To maximize damage with Javelin Toss, run away from the direction you throw the Javelin, or pop into Cougar Form and Pounce away.  As AP Nidalee, you are usually able to 1-shot or 2-shot a carry with Javelins from a long range.

It can be argued that maxing Javelin Toss first is a bad choice because of the "difficulty" to hit during lane phase, but that's up to the player you're laned against.  If you're not hitting them consistently, max E - Primal Surge / Swipe first.

Nidalee's biggest strength is her huge damage from her Cougar Form combo.  

While trading damage or poking, you can follow up with Pounce > Swipe > Takedown > run away for a significant amount of damage.  Against a less skilled opponent, you can even hit them with a Javelin after retreating.

I take Bushwhack at level 1 for brush control, gank-prevention, and so I have Pounce at level 6.  Since it's mostly a utility spell, max it last.

Though I only take one point at level 1, the level 1 damage on traps is fairly high.  Keep your bushes trapped constantly and ALWAYS get a few auto-attacks in if the enemy champion hits the trap if it's possible.  A few traps in the river + bushes can prevent early ganks, but you should still ward.

Pounce can be used to jump through/over a lot of terrain on Summoners Rift.

Here's a list of a lot of the places you can Pounce:

Top lane ridge between baron and blue team bush.

Baron Wall

Dragon Wall

Golem Buff Walls

Lizard Buff Walls

Wraith Wall @ River side

Double Golem Wall

Inner Side Lane Walls Between Inhib Turret and First Lane Turret

Wall in front of Wolves.

Using Pounce in combination with Flash can close huge gaps as well as create them.  Nidalee has HUGE kiting and escaping potential because of this.  Abuse it to no end.

Primal Surge used to be strong, but since nerfs, it's pretty underpowered.  The mana cost on CLGaming is wrong, so plan accordingly.

Mana Cost - 60/80/100/120/140 Mana

I choose to max heal second of my non-ult abilities for more sustain in lane, and there isn't a better option.  This should keep you on top in most lane matchups, as well as providing a decent attack speed increase for early game damage trading, harassing, and full-game pushing.    

Again, the mana cost is completely insane now, so try not to spam the hell out of your heal unless needed.  A few HP pots go a long way, and should be used before you spam heals, unless you have golem buff or full mana.

Everything I would've said about Cougar form is covered in HotshotGGs guide, so I'll just paste his, rather than just reword it.

It grants extra resistances so before large bursts of damage try to shift into form.

If you are being chased by enemies DON'T switch out to heal, even if you are 50HP. Always choose to pounce away.

Aspect of the cougar grants 20 move speed, always use that to your advantage when kiting and chasing your enemies.

When battling another champion it's almost always better to stay in cougar for the whole duration of the engage. Only use your human form abilities if you're able to kite during the cool down of your aspect.It grants extra resistances so before large bursts of damage try to shift into form.