Hey I'm Jack. I might not be the best player, as I'm only Silver III at the time I write this guide.
But I still hope that I can present my playstyle good and that some people maybe learn how to play with and against Nidalee :)
My LoLKing

Pros

+ Very good laning phase with bully potential + High mobility due to Cougar Form and Passive, especially in the Jungle + Bushwhack are a good replacement for Wards if you have none (still: Wards > Bushwhacks)

Cons

- High Mana costs - Needs Items to be usefull -> weak if behind - Gets outscaled in lategame, so you have to snowball properly if ahead

Flash Flash is the best option to run on Nidalee because you have a second escape or gapcloser, depending on how you use it. You should tend to use it more for defensive purposes than for offensive ones, because it's on a very high Cooldown and you shouldn't be picking up Summoner's Insight from Masteries, but ofc you can get the Enchantment: Distortion.

Ignite Ignite is a good option if you want to have more killing potential or lane pressure. You should consider taking Ignite if you are up against a Champion who can heal him/her very effectively, the best example being Dr. Mundo.

Teleport Teleport is especially good in Teamrankeds but can also find its place SoloQ. Teleport is a very good spell if you want to roam and make plays everywhere on the map. I just don't like it because I always forget it but that's another story.

I use this most of the time because it provides good base damage and resistances.
You can swap the Armor Seals for scaling Armor or scaling/flat HP if against an AP Lane and you can swap the MR Glyphs for scaling MR or scaling/flat CDR if up against an AD lane.

Passive: Prowl/Hunt Prowl: Moving through brush increases Nidalee's movement speed by 10% for 2 seconds, which increases to 30% if she is moving toward a visible enemy champion within 5500-range. Hunt: Hitting enemy champions with Javelin Toss and Bushwhack will mark them as Hunted for 4 seconds, revealing them. While moving toward Hunted, Nidalee gains Prowl's bonus movement speed and ignores unit collision. Additionally, Nidalee's first use of each of her Aspect of the Cougar abilities against Hunted will have an empowered effect.

Human Form Q: Javelin Toss ACTIVE: Nidalee hurls her javelin forward in a line, dealing magic damage to the first enemy it hits. After having traveled 525 units, the javelin's damage will be increase by 2% per 7.75 units traveled, capping at 200% bonus damage (300% total damage) after traveling 1300 units.

Cougar Form Q: Takedown ACTIVE: Nidalee's next attack gains 75-range (200 total range) and is modified to deal magical damage. Despite dealing magic damage, Takedown will apply life steal. The damage is increased by 1.5% for every 1% of the target's missing health, for a maximum of 150% bonus damage (250% total damage). HUNT BONUS: If the target is marked as Hunted, Takedown deals 33% increased damage.

Human Form W: Bushwhack ACTIVE: Nidalee lays a trap at a target location that arms after a very brief delay. When sprung by an enemy unit, they will take magic damage plus a percentage of their current health over 4 seconds (max. damage 150 vs. monsters). Traps last for 2 minutes.

Cougar Form W: Pounce ACTIVE: Nidalee lunges a fixed 375 units in the direction of the cursor, dealing magic damage to enemies within a 75-radius area around her landing point. Killing a unit will reduced Pounce's current cooldown to 1 second if it's currently higher. HUNT BONUS: If the cursor is near an enemy marked as Hunted within 750-range, Pounce becomes a targeted dash.

Human Form E: Primal Surge ACTIVE: Nidalee evokes the spirit of the cougar, healing a target allied champion or herself and granting them bonus attack speed for 7 seconds.

Cougar Form E: Swipe ACTIVE: Nidalee claws at enemies in a 180-degree cone in the direction of the cursor, dealing magic damage to all enemies hit. HUNT BONUS: If Swipe hits an enemy marked as Hunted, Pounce's current cooldown is reduced to 1 second if it's currently higher.

Ultimate: Aspect of the Cougar ACTIVE: Nidalee transforms herself into a cougar, gaining a new set of abilities and melee attacks (125 range). Triggering The Hunt resets Aspect of the Cougar's cooldown if Nidalee is in Human Form.

You want to always skill your W at level 1 because you can set Bushwhack to protect your Jungle from invades and to set some pressure onto the enemy Jungle or maybe even delay the buff attack by threatening the enemy Jungler and Toplaner. You also have your Pounce if the enemy is invading. Just make sure to stay as safe as possible so you won't end up dying for first blood.

Coming into the lane you want to shove the wave as quickly as possible to gain the level-advantage and harass the enemy laner by autoattacking them. A good way to do this is by siting in the brushes and running towards the laner, autoattack him and run back into the brush to remove the creep aggro. As soon as you hit level 2 quickly put a point into your Q and try to take the initiative. If you harassed the enemy you can now walk up to them, autoattack and Javelin Toss right after, go into cougar form, jump onto him and hit him with Takedown. If you took Ignite over Teleport add it into your combo and you'll at least burn his Flash and/or most likely gonna kill him. Add Swipe into your rotation at lvl 3 and keep harassing the enemy to bully your lane and gain the advantage. Of course this doesn't work against every enemy but most melees should have a hard time laning against you.