I apologize for my laziness and overall guide quality, but I left detailed notes for each of the build options. Make sure to read them to understand the logic behind some of the choices. As time goes on I'll come back and add more sections to this guide with better explanations.

Intro: Kiting Monsters
Just a quick run-down on kiting monsters with Shyvana, in order to minimize your damage taken early in the jungling phase. You can skip this if you're familiar with kiting strategies already, as Shyvana has a very straight-forward kiting strategy.

Shyvana's W and E have roughly the same duration and cooldown, and you'll usually use E and W immediately on every jungle monster, so the logic behind Shyvana's kiting strategy is straightfoward:

• At the buff monsters, wait until both your E and your W wear off (for E, the glowing symbol on the monster will disappear, and for W, the flames at your feet will dissipate) then, immediately after your next basic attack, run away a little bit. The idea here is to avoid a basic attack or two while your E and W come off cooldown, in order to maximize the ratio of damage dealt to damage taken.
  
• Treat Gromp the same way you treat the buff monsters.
  
• At raptors, basic attack of the raptors at the edge of the camp and take a couple steps backward. You'll notice all the raptors line up in an attempt to come attack you, so this sets up a nice E for you that will hit all of them. After you E (and W), run to the mama raptor and dance around her. That is to say: basic attack then take a step around her, then basic attack again, then take a step around her again. As you circle her, it'll make it harder for the little ones to hit you, saving you a surprising amount of health.
  
• At Krugs, the strategy is similar to the raptors, except that they're already in a line for your E. Stand next to the big one and E them both, then dance around the big guy so the little one can't hit you as easily.
  
• There's no real strategy to wolves. Just make sure your E hits all 3 at the beginning.

As a side note: I usually opt to wait for my E to come off cooldown before I start Gromp. Shyvana's jungle timing really only needs to coincide with dragon timing, so the extra few seconds don't amount to much.

Starting Blue (Blue Team)
When starting at Blue Buff on Blue Team I normally don't contest the Baron-side Scuttle unless I know for sure that the enemy jungler started at their Blue Buff. As Shyvana, I care much more about the dragon-side Scuttle, so I'd rather get to that side of the jungle as fast as possible. If you do want to contest the Baron-side Scuttle, you should do so right after taking Blue Buff.

With Hunter's Machete
Blue Buff --> Gromp + Smite --> Red Buff --> contest Scuttle (+ optional Smite) --> Raptors + Smite (if not blown on Scuttle) --> Wolves (if you didn't win Scuttle) --> Krugs + Smite

If you can secure scuttle, this route has the best timing for securing the first dragon on blue team for a couple reasons:

• By skipping wolves you can recall, buy, and run to dragon right as it spawns.
• Scuttle vision will still exist by the time you get to the dragon pit.

With Hunter's Talisman
Blue Buff --> Gromp + Smite --> Wolves --> Red Buff --> contest Scuttle (+ optional Smite) --> Raptors --> Krugs + Smite (if not blown on Scuttle)

Starting Red (Blue Team)
With Hunter's Machete or Hunter's Talisman
Red Buff --> contest Scuttle (+ optional Smite) --> Krugs + Smite (if not blown on Scuttle) --> Raptors + Smite --> Wolves --> Blue Buff --> contest Scuttle --> Gromp

Starting Blue (Red Team)
With Hunter's Machete
Blue Buff --> contest Scuttle (+ optional Smite) --> Gromp + Smite (if not blown on Scuttle) --> Red Buff --> contest Scuttle (+ optional Smite) --> Raptors + Smite (if not blown on Scuttle) --> Wolves (if you didn't win a Scuttle) --> Krugs + Smite

With Hunter's Talisman
Blue buff --> contest Scuttle (+ optional Smite) --> Gromp + Smite (if not blown on Scuttle) --> Wolves + Smite --> Red Buff --> contest Scuttle (only if first Scuttle was lost) --> Raptors --> Krugs + Smite

This is my preferred start on Red Team.

Starting Red (Red Team)
When starting at Red Buff on Red Team I normally don't contest the Baron-side Scuttle unless I know for sure that the enemy jungler started at their Red Buff. As Shyvana, I care much more about the dragon-side Scuttle, so I'd rather get to that side of the jungle as fast as possible. If you do want to contest the Baron-side Scuttle, you should do so right after taking Red Buff.

With Hunter's Machete
Red Buff --> Krugs + Smite --> Raptors + Smite --> Blue Buff --> contest Scuttle (+ optional Smite) --> Wolves --> Gromp + Smite (if not blown on Scuttle)

With Hunter's Talisman
Red Buff --> Krugs + Smite --> Raptors --> Wolves --> Blue Buff + Smite --> contest Scuttle --> Gromp

No Leash Path
Note: take Hunter's Talisman for this path and start with W instead of E
Raptors --> Krugs + Smite --> Red Buff --> contest Scuttle --> Blue side in any order...

Tips for Taking Dragons

• Ask mid to ward the dragon side of their lane, so you know if the enemy mid is coming. Ideally, they'd ward the upper-right bush if on blue team, or the bush behind red buff if on red team. If your mid doesn't ward, you should prioritize these spots for your ward.
• Ward the enemy blue buff bush (if on blue team) or tri-bush (near bot, if on red team).
• Place a control ward near the dragon pit.
• Have control of the dragon-side Rift Scuttle.
• Wait until right after the enemy's Baron-side buff has spawned. Conversely, don't attempt to take dragon right after the enemy's dragon-side buff has spawned.

Dragonborn - This is what allows you to 1v1 any dragon at level 4.
Twin Bite - Try to use Q immediately after a basic attack lands for optimal DPS.
Burnout - Don't be afraid to use your W to run away. Other than that, it's very straight-foward.
Flame Breath - Be careful not to waste Flame Breath; if you know you're about to be hard CC'd, you should hold off until you're actually able to attack and take full advantage of the benefits of your E. Also note that, in dragon form, this ability is a small nuke, so don't be afraid to clean up a kill with it, or just drop it for some damage if you're not close enough to contribute to a kill.
Dragon's Descent - The knockback on this ability is relative small, so don't be afraid to use it to catch up to an enemy, even if you have to activate it in a sub-optimal direction. It will disrupt enemy champions, making this useful for taking out those pesky channeling ultimates.