Prussia is probably the most battle-intensive campaign in ETW. A small Protestant nation surrounded by angry Catholics (and angry Protestants), Prussia has few historical friends and far more historical enemies. However, this is offset by Prussia's superlative line infantry and its generally excellent roster of units to draw from, as well as great bonuses at the beginning such as a decent monarch, an excellent cabinet, plug bayonet already researched, an upgraded school, and a great gentleman.
Prussia has arguably the game's best line infantry, with reloading skill and accuracy most other nations can only dream of. They have slightly inferior melee skills to the very best of their opponents in this regard (Russia and Britain), but this is more than offset by their early access to bayonet technology. In addition, Prussia has access to a very generous pool of elite alternatives: 15 superior line infantry and 6 life guards (superior versions of regular guards!). On even hard mode and above, line infantry are easily the best early game choice to recruit, due to their excellence and their relevance in the late game.
In addition to being unusually excellent, Prussian Line Infantry are also initially very cheap, due to a combination of good ruler traits, the Absolute Monarchy government type, and a phenomenally good war minister.
Prussian grenadiers are nothing special (discounting their interesting uniforms): heavy troops that are most devastating in the early game but still retain usefulness in the late game. Given the excellent capabilities of Prussia's regular line infantry, though, grenadiers hold perhaps a smaller place in this nation's army than those of some others.
In addition to the regular lineup of cavalry most European factions have, Prussia has access to a handful of excellent light cavalry with the DLCs: Bosniaks (superior Uhlans), Death's Head Hussars (light cavalry with very high defensive values), and 2nd Hussars (superior Hussars). Death's Head Hussars are of greater interest here as they are accessible very early on in the game. Unfortunately, all the DLC units are limited to one regiment.
Prussia has some great options regarding light infantry. Prussian Jaegers, their equivalents for Riflemen, can be trained twice as quickly as their counterparts with the only drawback being the fact that they may only be recruited in Brandenburg and East Prussia; Frei-korps are oversize, superior light infantry regiments that can also handle themselves in a melee.
Prussia has access to the full standard roster of European artillery, with no strengths or weaknesses to speak of.
Given the excellent capabilities of both line and light infantry that Prussia has access to, they benefit perhaps less from integrating artillery into infantry lines, and instead use artillery as its own force. In that capacity horse artillery might be better suited to the Prussian army, although playstyle comes down to the individual.
Same as the above, although a viable navy may be difficult to build given Prussia's relatively landlocked position and the hostility of its sea-faring neighbor, Sweden.
At the beginning, Prussia is poised to enter a bloody, protracted three way war with Poland-Lithuania and Austria. While it is possible for Prussia to come out on top, success is not guaranteed and the going will be definitely slow. I found that it is best instead to seek an alliance with Poland-Lithuania. They may arbitrarily break it once on the first turn, but just ask them for an alliance again and they’ll join (hopefully permanently this time). An alliance with Poland-Lithuania makes East Prussia virtually impregnable, and frees up your armies for some of your other problems.
So, where to attack? Courland, Saxony, and Poland-Lithuania are obviously out of the question with this strategy. In fact, allying with Saxony and trading with it is both secure and profitable. Attacking Hannover is also a bad idea; it’s one of the only factions that is friendly with Prussia in the region, and it’s allied to Great Britain, which would destroy all possible ambitions for trade and colonies for a very long time should it declare war on Prussia. This leaves Sweden, Denmark, and Austria. Denmark, with its weak armies and excellent lands, is the clear choice.
Disband pikemen and the demi-cannons: pikemen only serve to eat up upkeep, and line infantry are pretty much better in every way, while I’ve never found demi-cannons very good: if lucky, they might score a good hit or two against cavalry, but that’s pretty much it. Build a brig for transport, and take the Brandenburg army to Copenhagen and crush it. Hopefully the attack will be coordinated nicely with Sweden, which almost always takes Norway and destroys Denmark’s fleet in the first turn or two. Denmark should now be isolated to Iceland, and is no longer a threat.
Sweden shortly will declare war on you, but this is a blessing in disguise: with some reinforcements, your army should be superior to the Swedish army by this point, and Scandinavia is rich with resources and is very hard for Sweden to reinforce once it loses Stockholm. If you can, use whatever brigs you have to capture the Swedish fleet, take its 5th rates, and sell everything else, giving you dominance in the Baltic Sea. Take Norway first, which should only require a small garrison to keep unrest down, and then march on Stockholm and knock it out; immediately request for peace and Sweden should accept. Now Sweden should be isolated to Finland, St. Petersburg, and Estonia, where it should hopefully survive losing its capital and tie down Russia for some time until you are ready to take on Moscow.
While doing all this in the north, Austria will almost certainly declare war on you in the south. Raise a new army in Brandenburg, and use the army in East Prussia together with this army to march into the Hapsburg domain, annexing Silesia, Bohemia & Moravia, and Austria itself. From here it’s not necessarily a bad idea to just take over the rest of the Austrian empire, but I prefer to turn west and unify Germany. After digesting newly acquired lands, declare war on Bavaria, then immediately ask for peace with Austria, which should accept. This leaves Bavaria isolated; march in and destroy it. The defeat of Bavaria opens the way for Wurttemburg, Westphalia, and eventually France…
It is best to attack an ally of Hannover in order to provoke it to declare war on Prussia, rather than declaring war on it and potentially causing Great Britain to join the fray. Sometimes, Hannover declares war on Saxony, which gives a convenient excuse to obliterate it (handily grabbing Saxony to boot after Hannover annexes it).
By the time the smaller German states have been annexed, Prussia should be the undisputed leader in central Europe, with wealthy lands, lots of research, and a large, capable army. This leaves three broad problems: France, Poland-Lithuania, and Russia. France could be encouraged to trade its province of Alsace-Lorraine for something valuable, or you could invade it, which would probably cause Spain to declare war on you. Taking these factions out won’t be easy, as they have large armies, and their lands are almost completely Catholic.
On the other side, Poland-Lithuania has benefited from the destruction of Austria nearly as much as you did. At this point, it probably has very large armies, and has significant land holdings, making it difficult to quickly destroy. Diplomatically at least, it poses less of a problem: declaring war on Courland (which should hopefully still alive at this point) forces Poland-Lithuania to break its alliance with you. Watch out for Russia, which has probably spent the entirety of the campaign up until this point growing rich and powerful improving its lands and training armies.
One more thing to consider: by this point the Maratha Confederacy should have united India, and it’ll often declare war on you, presumably out of boredom. Expect ships carrying massive armies. With a good navy, you can hopefully intercept and sink them before they unload their cargo and cause something ugly to happen.