Kayn is weak early game as he does not get a passive until he has form, which is why he does not usually take drag until after unless it is really safe. Red orbs are gained from using abilities on any melee champion, and blue orbs are gained from using abilities on any ranged champion. This means you need to prioritise your ganks and pathing for which lane will give you the orbs you want.
Shadow assassins passive is that 40% of the damage dealt to champions is dealt as bonus magic damage for the first 3 seconds of battle, and is refreshed from the R.
The W becomes longer and no longer affects Kayn's movement, and The E cooldown is greatly reduced.
Rhaast passive is that he heals for about 40% of all damage dealt to champions from his abilities.
the Q cooldown is slightly reduced, his W also now uplifts enemies, and his R now heals him for a percentage of the targets maximum HP

When planning to go Shadow Assassin, you get 3 or four points into Q for clearance speed, then max W because of the excellent burst damage it provides.

When planning to go Rhaast, you max Q as it has a lower cooldown and heals you, than max W. Very simple.
W and R is Blue Kayn bread and butter, and Q and R is Read Kayn bread and butter.

1. If the enemy team has 2 tanks and 3 squishy champions, go Shadow Assassin and goredrinker(preferably if they are ranged).
2. If the enemy team has 2 or less tanks and 2 or more ranged champions, go Shadow Assassin into duskblade or prowlers claw.
3. If the enemy team has 2 tanks and a fighter, or 3 tanks, or 4 fighters, go Rhaast with goredrinker.
4. you should know which form to pick in champion select

Red buff -> E to Krugs -> E through to raptors -> q and w wolf camp, then smite the big wolf and attack it 3 times -> E through to blue buff and attack it 3 times, than Q to your left to finish the wolf camp that followed you -> auto attack grump and 1v2 with blue buff -> save smite for scuttle if you can, or smite blue buff, as it has more HP than grump,
This should be finished at 3:07, and you may have time for a gank, otherwise... just turn it away from scuttle.

Start Q at wolves -> raptors -> Red Buff -> krugs -> gank bot -> bot scuttle -> gank mid -> top scuttle.

As said by Karasmai, kayn's prorities are the following:
1. Farm
2. Form
3. Objectives
You want to balance farm and form for the first part of the game (form comes from ganking), and then play for objectives after form. Still try to get objectives before form, but it is not a top priority. Kayn is the fastest jungle clearer in the game and must utilise the early farm lead.
Before form, it is ok to turret dive if you have your R as you can easily escape. Moreover, even if you don't kill them, you are getting orbs for form. Kayn is one of the only champions that will always get something out of a gank. However, Kayn is a weak champion before form as he essentially has no passive. Look for dives when you have R, but not against fed champions, or ones with CC abilities.

Shadow Assassin requires lots of forward planning, and you need to try and get an early lead.
this is the reason for rushing Umbral Glave, as it is a small power spike that allows you to snowball.
you need to play very safe and only engage a team fight if the enemy has used their CC abilities, or if you have stopwatch. this is also one of the benefits of Goredrinker and Ravinous Hydra, as you deal an absurd amount of damage but are extremely difficult to kill.
you must be patient and pick off the enemy players, in a low risk high reward playstyle. try to play around your R
A good method is taking out the ADC in the backline, ending with R, and if you have duskblade, you will be invisible for an escape. But you can always flash, Q, and E instantly upon exiting the R to safely disengage from a team fight if feeling unsafe.
Remember, although Shadow Assassin is relatively safe due to the mobility, you still want to play him carefully to ensure you get a lead, and don't over estimate your strength. As in lore, you must play in the shadows, preying on fear.
lastly, as Shadow Assassin, you need to try and ignore tanks in the midgame or team fights until you get the item Grudge.

Rhaast is Simply an immortal teamfight machine, and works best in jungle team fights where there is small space for enemies to move.
The most important thing about Rhast is choosing and timing the R. You must not use it lightly as it can dictate the outcome of a team fight.
You need to save the R for low HP (maybe a third or a quarter) and ult the champion with the most HP. The purpose of the R is healing Rhast, not for damaging an enemy. stay in the enemy while your abilities go off cooldown then when you exit toward the enemy you want to kill. The R is a tool more than a weapon.
You will look for the ADC in a team fight and target them, but ult their support when on low HP, and resume killing the ACD after a HP reset. The only exemption with ulting a squishy champ is if they are alone, or just tried to flash away.
As Rhaast, your sole purpose is team fighting, and dragons allow you to do that more safely. You must prioritise dragons as Rhast as they will ensure your immortality. An early Formed Rhaast with dragon Soul is GG.

As Shadow Assassin, you usually want to enter a fight with Q, then W, as it is likely that they will try to dodge your Q or get distance, and it is harder for them to dodge your W. If possible, R the instant you cast W, and end it shortly after the W finishes for maximum one trick potential and safety, as you were only in that fight during your Q. If you are behind and struggling to get kills, it may be worth while to save the E for an escape.
In team fights, you want to move forwards, W their most fed player and Q backwards instantaneously. this is an effective poke that puts you at a safe distance afterwards. After a few of these pokes, you can jump straight into the fight with your R, and make sure you stay in the enemy for the maximum duration, as it allows your abilities to go off cooldown, and allows your team to come and help you. Exit the R, flash, Q, and E at the same time to blink out of the team fight. This is an effective combo because you can go into a 1v5 safely, and escape safely with the life of their most fed player. When your team follows they will then win the team fight.

Rhaast does not really have combos, he just wants to ability spam due to the 40% spell vamp his passive provides, and he looks for the perfect R. However, a small combo is just using gore drinker and q at the same time. doing this in a team fight after landing a W will ensure a health bar reset.

In conclusion, Ravinous hydra is the consistent item through both forms as it is the best item in the game, no one else will say to get it, but trust me... you need it. Shadow Assassin likes picking off enemies and ulting straight away, whereas, Rhaast loves team fights and wants to save his R for the very end of his life. play more for farm on Shadow Assassin and carry a game, but play more for drags as Rhaast and play for your team. Kayn is a safe and fun champ that I can never get enough of, and I hope you guys love him too.