Shyvana into Garen is an annoying lane but Shyvana can actually win it when played right. First of all, I generally like to get some attack speed on rune pages with Shyvana. I usually go with Flat AD marks, Armor seals, Scaling MR glyphs and Attack speed Quints. I also some times switch over to a full Attack speed and Armor rune page depending on the match up. The masteries are usually between Fervor of Battle or Grasp of the undying, depending on the match up. Against some one like Garen, where he'll probably beat you on the tank side, I'd probably go Fervor of Battle but if you are facing a tank that doesn't deal annoying damage then usually Grasp.
As for items, I usually always get these items.
The Titanic Hydra is something I'm testing out recently and consider it a nice item for way later into the game, that's the item I think of as a Situational item and can be swapped out for anything like another Armor item if your team feels like it needs more tanks or you could replace it for another damage item like
.
Now how to actually fight Garen. Since Garen is annoying with his Silence which he usually will start off with at level 1, start off with your Burnout. You also should always be maxing it first and just casually farm with your auto attacks and wait for him to start trying to fight you. When you see him use his Q and start running towards you, Activate burnout and when he silences you, you can still do a lot of damage back to him...Hell...if he does it in front of your minion wave you could just ignore Garen and have the minions + your burnout make it a positive trade for you...but that all depends on the current situation with who has more minions etc.
Usually with Shyvana, she would be able to just run into the enemy top laner with Burnout and E>Q them and in most cases Shyvana wins against a lot of melee top laners. The only champions that personally gives me trouble when I play Shyvana is
due to either taking my AD away (trundle) or just completely ignoring my armor like Olaf. While I'm not saying that other top laners aren't a problem, because
can also be a problem and beat me but usually it's either they are just better than me or maybe I used the wrong runes/masteries since I switch between them a lot and could pick the wrong ones some times lol.
Also you mentioned W, Q, E in that order, I'm not sure if you use that combo in that order but you should generally use E before your Q since it actually gives you more damage in the end that way. The only other thing I could think of is to be careful of when you jump in on Garen. If you are the Villian then I'd avoid fighting him if you can until it switches over to some one else because the amount of extra damage he does is pretty high and hard to factor in when fighting him.
This is coming from Shyvana main that's been playing her Top since Early Season 3...Even back when she was considered bad and was considered a "troll" pick for top lane back then lol.