"Soldier, let's go, let's go, let's go! And for you my dear, I have sent you mail from here. Farewell, the trumpet calls, Soldiers - on the march."
- – V Put (Lets go), Soviet/Russian Drill Song
"I Officially declare this FUCKING Party started!"
- – Vostroyan Commander at the beginning of the Battle 
The Vostroyan Firstborn are Russian Streltsy/Cossacks IN SPESS. They use old weaponry they consider valuable - this being 40k, older is better anyway. They wear richly colored coats and large fur hats. If you remove the coat and look at some art for them, they actually look extremely similar to the pre-Heresy Imperial Army's soldiers, which sort of make sense considering their origins in the Heresy.
Vostroyans are renowned for marksmanship, given they are using weapons that are quite devastating when used with precision, this results in the predictable outcome of the enemy being butchered (see Cities of Death cover art). Vostroyans have some of the finest training in the galaxy when it comes to firing drills and urban warfare, rivaling the Mordian Iron Guard (and surpassing it in some opinions). They're the nobbiest of the nobs.
History and Culture
As the Traitor Legions made their beeline for Terra, the Loyalists sent out a call for anyone who could bear arms in defense of the Imperium. The Vostroyan officials declined, explaining that as an Industrial World, Vostroya would be more useful making weaponry for the Imperium than firing them. True or not, after the Traitors were routed, Vostroya was called to account for its actions. The planet was ultimately spared, and as a sign of repentance the Vostroyans swore to pledge their firstborn children to the Imperial Guard from then on.
Different regiments are kept with a steady flow of recruits due to the nature of the regiment's recruiting methods, thus their regiments have a tendency not to be completely destroyed and have enough veterans kicking around to show newbies the ropes before they are used as a toothpick by a Carnifex or something. It's also a tradition to serve in your family's particular regiment, i.e. you're likely to wind up in the same squad with your uncle, cousin, great-uncle or whoever. This serves multiple goals: first, you'll become part of the team faster; second - one is more likely to fight bravely and/or hold the line knowing not only his honor, but his family's would be screwed forever otherwise; third - most Vostroyan weapons are relics passed from one Guardsman to another, some of them since the time of Great Crusade, and being able to simply pick it up from your relative's cold hands simplifies the inheritance procedure greatly.
They like their moustaches. Their coming of age is sprouting a four foot long moustache and goatee combo, if male; if female, on your 16th birthday you take a deep breath and pop out breasts like a squeeze toy. Such is their affinity for the mustache that anyone with an aversion to body hair is treated as the most traitorous of scum. Vostroyans will always squat rather than sit to keep the snow out of their britches (leading to some tech-priests suggesting that they can just leave seats out of Vostroyan vehicles to save on material). In their downtime, they drink a lot Kvass made from the piss of Vostroyan urban potato bears. They are known to dance into battle to the beat of an obscure techno-music that the Mechanicus dubs “hard-base” like this: 
Typically they all look like so: 
Their home world, Vostroya, lies in the Segmentum Obscurus and is quite cold. Vostroya isn't a Forge World nor is it an Imperial Hive World, so it falls in between and gets the award of being an Industrial world. The Mechanicum still gets a fair bit of sway there though, being represented in their government and all, as they were found by Martians during their early exploration efforts, before the Great Crusade, and as such produce almost all wargear for their regiments, including Baneblade variants and some unique patterns, on-planet.
Vostroya isn't exactly a friendly place to live in, as its landscape is bleak and overpopulated, with plant and animal life almost extinct outside of human control. Because of this, the population of 9.3 billion is in a state of constant change over. This also means there is a plethora of firstborn children to fuel the Vostroyan Firstborns' ranks.
They should be on the look out for the birth of Space Peter the Great/Taras Bulba, whom the Imperium is in desperate need of.