First thing you need to do is assess whether or not you'll need to fight early, or if you'll have the space to farm. If you are going to need to 5-man early, Drums are usually a must, and you will probably need BKB. Consider SnY en lieu of BKB, as it gives you a bit of everything, most importantly HP and movespeed.
If you think you have space, Helm is core as a first item (after Aquila and Phase of course). You want to take their offlane tower asap, then rotate between flash farming the jungle and pushing out your lane, all while stacking ancients. You should be super farmed after a few ancient stacks. But don't forget to carry tp and help your team when you need to, Gyro is very strong early and it's a waste not to take advantage of this.
After this, itemisation depends on the game. If your team has some decent teamfight abilities, usually you want damage asap. Don't be afraid to rush MKB - you can get it really quickly and it will be easy teamwipes combined with some good lockdown (Void, Warlock, Enigma, etc.) If your team isn't composed like this (ideally it should be, Gyro is a teamfighting carry), consider Manta/SnY into Skadi. You'll need to be tankier and a bit more slippery to survive fights. Remember that BKB will need to be purchased at some stage in most games - never delay it for a different item if you need to win fights NOW.
It sounds weird and greedy, but I have had a lot of success in slower-paced games rushing both Butterfly and MKB. You can get both before 28 minutes if you're farming efficiently and taking your stacks, while making sure you help your team at pivotal moments to ensure you stay in the lead in terms of map control. Getting an early Demon Edge will also increase your farming speed tremendously, particularly when taking ancient stacks.
Ideally you don't want to go to late game - you want to be knocking on their rax at 30-35 minutes. But later items after making your selection from the core (Drums, SnY, Manta, BKB, MKB, Butterfly, Skadi) include Satanic, Linken's, Blink, BoTs, Abyssal and of course Divine Rapier.
On Dragon Lance: not a fan. Yes, Gyro often wants both BKB and Butterfly. But he usually wants to be in the thick of the fight; BKBing, ulting and hitting their entire team at once while they scramble. Lance is good on heroes who rely on kiting like Sniper and Drow - generally speaking Gyro doesn't. He enters the fight in a blitzkrieg at a core moment and wipes out at least 3 heroes. He only needs to attack one target; his Flak has great range already and is unaffected by Lance.
If you need to 5-man, the gold spent on Dragon Lance could have gone towards a faster BKB or SnY. And if it's a slow game, you could be buying farm acceleraters like Yasha or Demon Edge. In some situations, I'll concede it might be an ok pickup. It might be nice with Void as your offlaner, just to ensure you can hit them if it's not a well-placed Chronosphere, for instance.
I apologise for my post being all over the place if it reads that way - I really need to stop writing long posts on mobile. :D