I think I've talked about this here, but I'm happy to go into more detail. When it comes to regional specialties of warfare, in some cases it's easier to identify trends than others, but I'll try my best. (I'm also going to tag in @warsofasoiaf, because I'm pretty sure we've talked about it this quite a bit over the years.)
Keep in mind that we're talking about minor variations in Westerosi armies that all tend to have a mix of knights, infantrymen, and archers working together in a late medieval paradigm of warfare heavily based on the Wars of the Roses.
The North:
The North's speciality doesn't seem to be a particular kind of military unit, but rather a kind of unique elan: when we consider the campaigns of Roddy the Ruin (the Fishfeed, the Butcher's Ball, First Tumbleton, and Robb Stark (Whispering Wood, the Battle of the Camps, Oxcross), one of the common trends is that Northern armies often employ a strategy of unexpected reckless charges that rely on aggression and shock to panic the enemy.
Usually, this leads to an emphasis on Northern cavalry as the core of the Northern army, but we also see Northern infantry do the same thing at the Battle of the Green Fork. There are some tactical differences - Roddy tended to favor frontal assaults at the risk of heavy casualties, while Robb Stark favored the use of maneuver and ambush.
Here, I think the North's ferocious if somewhat ill-disciplined approach is evocative of the Highland charge and the way it absolutely devastated much larger, European-trained professional armies until the Battle of Culloden, when changes in bayonet technology and a shift in bayonet tactics to solve the problem of the Highland targe.
The Vale:
We haven't seen the army of the Vale in operation, so we can't really talk about a specialty. However, it's worth noting that repeatedly in the text, the Vale's army is referred to as "the knights of the Vale," so that fits with their general tendency to cultural traditionalism.
The Ironborn:
Essentially equivalent to historical Vikingers, they're very good at naval warfare, marine operations, and ambushes on relatively undefended targets. They make up for the generally smaller size of their ships through the use of misdirection and distraction.
On land, they fight in (somewhat outdated) infantry shieldwalls and are heavily hampered by their total lack of cavalry, and their relative lack of archers. Moreover, they have no experience of fighting cavalry and tend to break in the face of a cavalry charge. They're pretty poor fighters on land, to be honest.
The Riverlands:
As we see both in the Dance of the Dragons and the War of Five Kings, the Riverlands has a historic tendency of constantly being invaded by all its many neighbors. While they often come out on the losing end, they tend to be more successful when they eschew conventional medieval tactics and instead focus on guerilla warfare or defensive setpiece battles that emphasize the use of rivers as natural barriers.
The Westerlands:
Because they have the most money - which is quite reminiscent of the Burgundians - the Lannisters can field multiple large armies with lots of extremely well-trained and equipped soldiers: their melee infantry aren't spearmen but well-armored men-at-arms and well-disciplined pikemen, they have a lot of knights. However, Tywin also has a tendency to use various less-disciplined fighters - the mountain clans, free riders, and some very questionable choices in mercenary companies - as expendable raiders.
The Reach:
As the most populous region in Westeros, the Reach has the advantage of massive numbers - it's full army is 100,000 strong - which gives it the option of attacking from all directions in force, essentially have the luxury of not having to choose where to spend its resources.
Moreover, the Reach is known for its knights - although as noted in my coverage of Aco*k, there's something very weird going on with the numbers of the Reach's cavalry, because it seems like it only has 20,000 knights, which seems rather low.
The Stormlands:
The Stormlands are known for their marcher lords, whose men are the best longbowmen in Westeros. They're also known for their castles, so I imagine the Storrmlands are particularly good at defensive siegecraft.
Dorne:
See here. The Dornish have their own entirely different tradition of warfare that bears almost no resemblance to the rest of Westeros, it's a mix of phalanx warfare, horse archers, and jinete light cavalry.
