Shyvana is one of the most versitile champs in the game.
Because of that you can play her into almost every draft which makes her a great champion to onetrick.
Once you get over her medium to weak early levels, she has great jungle clear and the ability to solo dragon very early very fast because of her passive Fury of the Dragonborn.

Right now hybrid bruiser shyvana is by far the best build but dependent on your draft an other builds also work just fine. In the end it just depends on what is fun for you to play.
If you descided on a general building path, what you build exactly really depends on the enemy team. Althought you can get away with just building the example build from the hybrid buiser build every game, its not ideal.
Experiment and build things that sounds fun.

• never waste time, take your camps when they are up, move around the map with intention!!!
• try to track the enemy jungler, which side does he start on which of his camps are up?
• dont over gank, especially in lower elos every bad thing that happens is junglers fault, dont let your tilted laners get to you you are not their babysitter!
• keep track of the laner states of the lanes you are pathing towards

In general you should always try to ful clear meaning you get all your camps on your first clear.
After that you can try to get the scuttle crab if the enemy jungler is on the other side of the map or you can 1v1 him.
If you start red buff or blue buff depents on what the enemy jungler picked and where he paths and on your top and bot matchups.
For example if the enemy top laner is a lot stronger that your top in the early levels, he likely will push in his wave to try to back on wave 3 or 4.
If thats the case you should think about pathing top.
If the enemy jungle is a lot stronger you in the early game, for example when its an Olaf, you should try your hardest to avoid him.
If hes starting top you start bottom and the other way around.
Your two options of full clearing are:
blue buff→ gromp→ Greater Murk Wolf→ raptors→ red buff→ Ancient Krug and then look for Rift Scuttler or a gank
or
red buff→ Ancient Krug→ raptors→ Greater Murk Wolf→ blue buff→ gromp and then look for Rift Scuttler or a gank.

Shyvana has a very weak early game so your main goal in early game is to reach level 6 so you can contest the enemy jungle and gank your lanes.
To achive that as fast as possible you will full clear almost every game.
Keep in mind that you are very vulnerable to invades early so tell your team to help you guard your jungle before the first minion wave arrives and ward your jungle if you suspect the enemy jungler will invade.
After your full clear you can try to look for ganks but dont forget that you have very bad ganks until you reach level 6, so if the enemy laner isnt overextending or your laner has good gank setup like Veigar Event Horizon, you can just recall.
As a jungler especially it is important not to waste time so recalling and full clearing again is not a bad choice.
The first dragon spawns at Minute 5.
Ony shyvana getting the first drakes is really important because of her passive Fury of the Dragonborn so if your mid and/or bot have prio(meaning they have pushed they wave under the enemy tower and therefor are free to roam) or you see your enemy jungler being on the top side(for example ganking your top laner or because he just took Voidgrub) you take dragon.
Outerwise you just farm and scale.

Mid game beginns around minute 13 to 15 when the first towers fall.
In most games your adc will move mid to farm and so your supp will be free to roam and help you.
If drake is up and your team has someone with Teleport they will be top and wo ever is left goes bot.
Playing for drake as shyvana is even more important than on other junglers because of her passive.
At this point your ganks arent that weak anymore so if dragon is not up then you just farm and look for skrimishes, ganks and dives.
Try to get creative with your R Dragon's Descent and use it for example to jump over raptors pit onto mid lane to catch someone sitting under tower of guard.
Your main objectives for mid game are to scale and take dragons.
Tradeing dragon for voidgrubs or rift herald is alway worth for shyvana.

Play around objectives. Use your E ind dragon form as a zoning tool like a control mage.
Especially if you are behind you can look for splitpushs. If you are running Nashor's Tooth or AD builds you can tear down towers pretty quick. If your splitpushing you can also take the enemies jungle caps to get a level atvantage.
Dont aram mid but look for teamfights if you think you´ll win.
In that case you can also try to bait on nash(act like you start Baron Nashor as a team to get the enemies to misposition). (More on teamfights in the teamfight section)

How you approch a team fight depends on you build and yours and the enemies team comb.
If you are the only frontliner you have,stay with your team poke with E and peel(doing stuff so they dont die) your carries. Because you have no real hard CC you have to do it by zoning of enemies with E and jumping onto them with R if they try to catch your carries.
If you arent the only frontliner you have a lot more freedom in your approach.
You can obviously use you R too dive onto the enemy back line if you see a good chance(if they waste important cooldowns, misposition, etc) but also as a tool to flank.
If you fight around dragon you can for exa,ple use it to jump over the dragon pit to get new creative angles onto them.