Ok so im a pretty good shyvana player, but im no god. I can just see the potential through the numbers and from experience on how to play her well. So i am going to make this short and sweet.

For items you will want as follows:

Devourers
Boots of swiftness
Titanic Hydra

WAIT STOP!!
this is the poin in the build where things can differ. until this point though you will want to be playing pasively, just farming as much as possible to stack your devourers and waiting until you are tanky to do consistant ganks or participate in team fights. So here are some items that can go well with shyvana, depending on what position the game is in.

Dead Man's plate- pretty standard, good damage and lots of tanky so pretty solid item for her at any point.

Zz'rot Portal- i love building this item because it can help you shove down lanes and just cant be ignored. buy this if you are behind on towers, or you want to push inhibitors solo often.

Randuin's Omen- Pretty good item, usually build early if an adc is fed.

Banshee's Vail- Good if an AP person is fed or a utility person is just ruining your day.

Thorn Mail- great for dueling but only buy this is there is some prevelant AD on their team.

BORC- great for dueling, but if you buy this make sure that you are no longer fufilling the MAIN role as tank. Try to build more items that give you tankieness without health when you get this to make the life steal more useful. Good if you want to be a mid game assasin.

Spirit Visage- Pretty standard and solid item, you can't really go wrong with it.

Sterack's Gage- Its decent on her. Nice shield for when u initiate fights as a shyvana should. only build this if you see yourself as a bit ahead and DO NOT build it 4th item.

Sunfire Cape- Pretty solid item like spirit visage. Lots of good things about it.

Last Whisper- Peronally I think this could go well on shyvana, but i don't see it fitting in to her build. If i were to get another offensive item for her i would probably get borc for the life steal but im not saying it isnt viable.

Trinity Force- Again, i like to build her more tanky than this, but if u want to play her as a mid game assasin then go for it.

Maw of the Malimortious- Good for the shield and for the assasin like play stile of shyvana, but not good to just throw into the tanky build.

Frozen Mallet- Shyvana can't be kited that well, but if you find that if somehow you are being kited a lot, this can be viable.

Warmogs Armor- If there are a lot of skirmishes with light casualties in each of them due to everyone disengaging a lot because they are little *******, (or just smart whateves) this can be nice to keep your health up between fights to have a constant presence.

Just aboutm no matter what, shyvana has a weak early game. once u get your titanic hydra you start to come online but until then try to stay in the jungle and be religious with your vision. remember that if you no longer need your w to farm, save it early game to help you escape. Max Q then W then E. In a team fight ure your ult then just basically roll your face on the keyboard. try to get to the backline and hit everyone with AoE on the way. In a 1v1, use your E early and try to get those auto resets. ANyways, thats pretty much it. Her counters are just early game champs. She can't be kited that easily. Good Luck.