Greetings everyone! My name is Tennokage and I am here to teach you how to play Shyvana!
I am a rank 7 Shyv, and I have been playing her since I first started, as she is listed for beginner champs and her ulti hooked me. I mean, who doesn't wanna turn into a f***ing dragon, cmon?!
Shyvana is very similar to Master Yi in the way that she power farms until she hits a few items, then is a hard carry. Due to lack of cc or any escapes/gap-closers (outside of her ultimate) she is very kite-able and easy to chase down if a gank or engage goes poorly.
Make sure that when you go in for an engage, your team is available to follow up!

Shyv is a great power farmer in the jungle, however she tends to get fairly low while farming camps, especially since the season 7 jungle update. Since Runes Reforged no longer allows us to really prioritize armor to help clear the jungle, make sure you ward a lot to be prepared for any attempted invades! After the first clear you will not get very low anymore and you will not have to worry as much.

There are many rune options viable for Shyvana in the new Runes update. I think in general the Domination skill tree is excellent for junglers and as such it is a viable tree for Shyvana. However, Press the Attack is a very strong rune on Shyvana, and because of that I prefer to follow the precision skill tree.

Precision:

Your keystone should be press the attack. Shyvana is heavily auto-attack dependent, and so it is easy to stack. In addition, her Q counts as two auto-attacks, so it puts 2 stacks on itself! Her damage is pretty ridiculous when you stack her E and Press the Attack's bonus damage. Conqueror is a viable alternative if you wanted to use something else under the precision tree, however I find it to be less reliable on junglers,since it is hard to proc while ganking.

Minor Runes:
Triumph: Hands down, the best option. It will save you in teamfights, tower dives, and in dueling situations. You are an excellent duelist, and triumph can be the difference between being shut down and getting a penta.

Legend: Alacrity: I choose Alacrity for the bonus attack speed, because I prioritize my speed. However, tenacity and bloodline are good options, so feel free to choose any of these!

Coup De Grace: IMHO the best option here. You are very strong and deal a lot of damage, so in most cases champions will be low when you are full hp, rather than the other way around. Last stand is only viable if you are a bad Shyvana and get low because you made dumb decisions. If you constantly find yourself in low hp situations, then by all means try it out. But seriously Coup De Grace is too good.

Secondary options:
Anything really. Its really up to you if you want tanky stats from Resolve or if you want Dmg from the others, or even utility. I personally prefer to use domination for zombie ward lol.


Domination:
I would say that you should prioritize press the attack on Shyvana, but if you wanted to spice it up Dark Harvest or Hail of Blades are decent alternatives.
Lets look at the benefits of Dark Harvest for a moment.
Dark Harvest allows you to collect souls from minions/monsters/champion kills that stack over the course of the game. After collecting a soul, you become "Soul Charged" for a predetermined period of time (20 secs before 150 stacks, and 5 mins after 150 stacks). While Soul Charged, your next attack will do bonus damage based off of how many souls you have collected.
Shyvana is a power-farming jungler, which means that she can just farm the whole early-mid game and stack this up. In addition, once Shyvana has used her ultimate, the most efficient way to get it back up again is to leave the fight and farm a camp to boost her fury back up. This camp would reset her soul charges and allow her to do the bonus damage once more, while also refueling her ultimate.

Hail of Blades on the other hand offer free attack speed at early levels, and without needing to wait for a few seconds for it to proc (like with Lethal Tempo), it is a very useful dueling and ganking tool. It also allows you to bypass the attack speed limit and therefore is pretty good late game too. Combined with a low cooldown, I honestly believe this to be the best alternative to Press the Attack on Shyvana. Precision offers the best stats for Shyvana, but Hail of Blades is pretty fun to try out.

Now for the minor runes:

Sudden Impact- To be honest the other two aren't that great on Shyvana. At least Sudden Impact gives a little burst of lethality and magic pen after the ultimate.

Zombie Ward- I think this rune is really too good not to have on a jungler, who carries a heavy warding role. Ghost Poro is ok, but if you stand in a bush for too long it will move to that bush, which makes it not as useful in my opinion.

Relentless Hunter- MS is always good on Junglers, who need to get across the map as quickly as possible. Shyvana doesn't really use abilities for damage, so permanent Spell Vamp isn't really going to help her. She also doesn't really use item actives too often to need the buff to those more than Hunter.

The secondary tree is where it is really up to your preference. If you want bonus attack speed on her you could go for Alacrity and Triumph under the precision Spell tree.
If you wanted to build Shyv a little more tanky, you could go for Demolish and Overgrowth under the Resolve tree. It's all up to you!

JUNGLE SHYV:

Start W, then E, then Q and max in that order; of course put a point in your R whenever available.
This allows for the most effective clearing of jungle creeps and ganks.

When playing jungle Shyv, always start bot!
It is just an easier start, with two people leashing rather than one.

STARTING RED:

Start red, then wolves, then blue, and maybe scuttle and gromp if you feel like it. Then gank or take krugs/chix.

STARTING BLUE:

Start Blue, then gromp, then wolves, then red, take scuttle/gank or recall if no opportunities are available.

Since the changes to Scuttle make it a viable objective to contest, it may be worth prioritizing as early as level 2. If you think you can get away with it, starting red then going straight to scuttle is an excellent way to secure some free gold and exp, as well as vision for your team. Alternatively, if you start blue, go to Gromp second then scuttle. Just keep in mind that this may put you a little behind on your clear, and also allow the enemy jungler to guess where you are if he sees the vision placed in the river.

When playing jungle Shyv, I try to get a full clear of the jungle before ganking. This gives me time to get smite, as well as my boots,and a control ward. Maybe you might be able to get a dagger as well. When ganking, look and see which lanes need help. If they are all fine, maybe take red again and then go bot, or counter jungle instead.

You should be able to take dragon by level four, provided you have a hunters potion with full 5 stacks, both smites available, and at least one dagger.

By level 7 you should be able to solo rift herald, provided you have hunters potion with 5 stacks, a recurve bow, and both smites.

Usually better to wait until you know where the enemy jungler is before you take an epic monster like rift or drake early, purely because you will be left VERY LOW and they may potentially steal your prey as well as get a kill on you! BE CAREFUL AND WARD! On the bright side, your passive gives you 20 percent bonus damage to drakes, and so the enemy jungler probably wont suspect you to be at drake at lvl 4 because they couldnt take drake probably to lvl 6 and above. Most dont realize how much that passive can help you take key objectives early!

With Shyvana, you want to be in the front lines of all teamfights! Soak up some damage and deal a lot of damage, as you will be a very tanky high dps fighter!
Ulti in, use your AOE Flame breath (e), then use your W and Q to finish the work. If you ulti in and realize that you effed up and need your team, ghost out of there (provided you took ghost and there isnt any hard cc).

JUNGLE SHYVANA:

Pros:

Fast and efficient Jungle clear!
Does well against most junglers!
Can do well even when behind!
Does heavy damage when ganking!
Good Gapcloser/Escape with ulti!
Ulti is reset quickly by farming! (can farm btw ganks to reset!)
Can Gank pre-lvl 6, although not as great!
Excellent at counter-jungling quickly

Cons:

Can get low while farming
Has no cc for ganks
Much less damage without ulti up
Can be slow and unable to catch fleeing enemies
Harder to contest scuttle without any cc

And there you have it, Tenno's guide to Shyvana!
She is a very fun and versatile champion to play, and I hop you guys enjoy playing her out on the rift!