Since gromp no longer gives poison damage from smite, start with krugs, to raptors, then scuttle crab for health regen and vision. Ward red to save for later. Move to murk wolves, to gromp, to other scuttle crab if there and ward blue (or have an ally ward blue for you). (*side note: this order can be reversed if you wish*) Go back, get jungle item and boots, then do a full clear in this order: Gromp, blue, wolves, chickens, krugs, then red. This will give you lvl 6 as well as a LOT of gold quickly. This puts you in prime position to either gank mid and/or bot lane(s) to secure the first dragon. After that, go back, finish jungle item and upgrade boots, then just farm up the jungle until another gank is possible. At this point, you'll be far enough ahead to roam everywhere.

G = gromp
K = krugs
C = chickens
W = wolves
S = scuttle

TL;DR jg clear: (G,W,S,R,K,S, back , G,Blue,W,R,K,Red,S) red buff last so last longer for ganks.

Depending on how far behind the team is, don't join then for it'll cause you to die, so just split push hard enough that the enemy team will HAVE to focus on you so your team can try to get back up to speed while the enemy team trys to chase you down. use ZZ'rot in the jungle bushes to keep the lane pushing in their base while the enemy team chases you down without realizing the tower/inhib is being destroyed. This stategy works well if the enemy team REALLY wants to kill you.

Story Time
So one game, my team was behind +10 kills at ~30min, all of my team makes were on average lvl 1 3 4/8/4 or so while I was 4/0/0 lvl 16, almost full build. I thought at this point that I had to do something or the other team would win, so I went back to base, finished my last item (zz'rot), placed it on the enemy team's bush between wolves and the turret and then went (excuse my language) balls to the walls on the enemy team, causing both teams to get pentakills on each other. At this point, all that was left mid lane was the inhib turret, inhib, and two nexus turrets and we all had 60+ death timeers. While dead, my zz'rot portal destroyed the turret, inhib, and most of the nexus turrets. When we came back to life, we had pretty much won the game due to zz'rot portal and a giant cluster**** that killed everyone so noone could touch my portal.

TL;DR even if you're far behind in kills, you can still win with split-pushing objectives.

For max dps on champions or drag/baron, got E, W, then Q to deal 7.5% max hp magic damage + 300 physical damage from Q + AA (Q resets AA timer), and AoE burn from cinderhulk and W. While W is active, auto attacks gain flat damage +20% of bonus AD. Just make sure you stay close enough to the enemy for the AoE to burn the enemy, but far enough away to kite their auto attacks.

All in all, Shyvana deals magic damage, physical damage, and %max hp damage which can quickly kill enemies if they are foolish enough to challenge you head on.