I'm afraid this won't be a complete answer since my focus is firearms. u/parallellpain had an excellent post on Sengoku cavalry and might be able to explain Japanese tactics and organization a better than I can. It doesn't seem that the Koreans were known for good cavalry either except for the very northernmost provinces, though much of Korea was fairly ill-suited for cavalry maneuvers anyways. You also might be interested in u/bigbluepanda's post about the use of glaives and naginatas here.

The majority of Japanese troops at the start of the war were armed with bows or spears, but what tends to get the most attention from modern scholars and historical sources are the 20% or so who were armed with matchlock muskets. From the account of Korean Minister Yu Songyong:

In the 1592 invasion, everything was swept away. Within a fortnight or a month the cities and fortresses were lost, and everything in the eight directions had crumbled. Although it was [partly] due to there having been a century of peace and the people not being familiar with warfare that this happened, it was really because the Japanese had the use of muskets that could reach beyond several hundred paces, that always pierced what they struck, that came like the wind and the hail, and with which bows and arrows could not compare.

When our soldiers are lined up against the enemy ranks, our arrows do not reach the enemy while their musket balls rain down upon us.

The Japanese muskets were typically around three feet long with a caliber of about 15.8mm, but came in a variety of lengths and sizes including small guns designed for use from horseback, long-barreled guns designed for accuracy, and what were essentially hand-held cannons that weighed more than 65 lbs. The Koreans had European-inspired cannons as well as old-fashioned "victory guns" (essentially a small tube on a stick), but according to Songyong there was only one musket in the royal armory prior to the invasion. Ironically this weapon had been a gift from the Japanese ambassador a couple years earlier.

The famous Japanese katana didn't receive as much mention during the war as polearms, bows and muskets do. However in the few battles it was mentioned it reportedly gave the Japanese an advantage due to being longer and sharper than the swords used by the Koreans and Chinese. One of the more iconic Korean cavalry weapons was their battle flail, however they also used spears, glaives, the occasional trident, and bows. The Koreans had a very strong archery tradition before the invasion and seem to have been considered better archers than the Japanese or Chinese were. After the war began though the Koreans started making a significant effort to capture, import or produce their own muskets to counter the japanese. In 1595 King Sonjo was reportedly observing some of the new Korean musketeers practicing marksmanship and was so impressed by their skill that he presented them a gift of thirty horses.

The first invasion in 1592 caught Korea unprepared and made relatively rapid progress, defeating armies raised against them and capturing forts and towns all the way to Pyongyang. In field battles the more experienced Japanese were typically able to outmaneuver the koreans, sometimes spitting their forces and coordinating an attack from two or three sides at once. According to King Sonjo the Japanese musketeers would be divided into three groups and "shoot alternately by moving forward and backward".

In sieges the Japanese preferred to use musketeers almost exclusively. Yu Songyong mentions small groups of japanese skirmishers being able to approach the fortification while avoiding enemy missiles and shoot defenders off the top of the walls. Ideally the japanese would try to find high ground overlooking the fort or build tall earthen mounds or "flying towers" with which they could shoot down on anyone hiding inside. After that the defenders would usually surrender.

Early on in the invasion a Japanese commander sent word back home stressing the need for more guns and ammunition.

Please arrange to send us guns and ammunition. There is absolutely no use for spears. It is vital that you arrange somehow to obtain a number of guns. Furthermore, you should certainly see to it that those persons departing [for Korea] understand this situation. The arrangements for guns should receive your closest attention.

The Ming didn't enter the war until after Pyongyang fell and were slowed down by the muddy terrain. When they reached the city they found the gates open and the city apparently abandoned, but when their cavalry entered they were ambushed by Japanese soldiers hiding in buildings and soundly defeated.

Eventually the Ming artillery arrived and they drove the Japanese out of the city with a storm of cannonfire and explosive fire arrows. After this defeat they adopted a policy of avoiding set piece battles against the chinese who had superior cannons and cavalry. Instead they began focusing on ambushes and guerilla warfare using the mountainous terrain to their advantage. This turned into a war of attrition for both sides as defeats at sea prevented the Japanese from establishing stable supply lines and and the Koreans began guerilla activities of their own forcing the Japanese into peace talks.

The second invasion wound up playing out fairly similar to the first. Japanese forces advanced out of their occupied territories capturing forts, then the Ming intervened again and started driving them back while Korean troops attacked where they were weak and Yi Sunsin defeated their navy. Despite the hundreds of thousands of Japanese troops sent to korea they were never quite able to secure their occupied areas. According to one Japanese General, "Korea is a big country. If we move east, then we have to defend the west; if we attack to our left, then we are assailed on the right. Even if we had another ten years, the matter still might not be resolved." Eventually Hideyoshi died and the Japanese withdrew after losing large numbers of men.

Swope. "Crouching Tigers, Secret Weapons: Military Technology Employed during the Sino-Japanese-Korean War 1592-1598"

Andrade et al. "A Korean Military Revolution? Parallel Military Innovations in East Asia and Europe"

Kang. "Big Heads and Buddhist Demons: The Korean Musketry Revolution and the Northern Expeditions of 1654 and 1658"

Chase. Firearms: a Global History to 1700

Andrade. The Gunpowder Age