"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
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.
Vostroyans are experts at urban warfare and firing drills. So they are like the Mordian Iron Guard with less dress up bullshit.
IT'S NOT BULLSHIT, IT'S TRADITION YOU DICKNOGGIN. *BLAM* No infighting between Guardsmen! 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.
The Vostroyan Firstborn are renown for their 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). Their name comes from the tradition of the first-born child of every family (from the commoners to the nobles) serving in one of these regiments, and they bear the name with pride.
The reason for this tradition dates back to the Horus Heresy. As the Traitor Legions made their beeline for Terra, the Loyalists sent out a call for anyone who could bear arms in defence 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..
This idea follows further, being explained that the 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. Also it's something of 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. This begs the question - if this method works so well, why is nobody else doing it (or something similar)? The answer is because it's a logistical nightmare.
GLORY TO THE FIRST MAN TO DIE.
Fantasy Flight Games changed it a bit by adding that the first-born daughters often join as well, to cease the angry outburst from women who wish to don bearskin hats and speak like they're in Red October. Even if these women have less endurance and strength than the men, the Imperium is unlikely to care that female Guards cannot keep up with the men for more than a short while or that exhaustion will quickly get those women killed. As far as the Imperium is concerned, a soldier is meant to die anyway. So long as their deaths get their replacements a little closer to winning, no one gives a damn about their sex or chances of survival.
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. Presumably Vostroyans drink a lot and dance 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 sons 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.