I'd also like book recommendations or any illustrations showing this information accurately if possible. As I understand, their warfare was nothing like games and movies depict it and I'd love to read or see accurate depictions of how battles were fought.
The concept of a 'unit' does not really exist in the modern sense for most of the Sengoku period. Each daimyo has multiple vassals, who when called upon to war, will bring a certain number of men - samurai, ashigaru (yari, teppo, depending on the vassal), mounted horsemen/cavalry, and so on. The number varied depending on the vassal's relationship with their daimyo as well as how wealthy they were - usually about 20 or so (this would include the logistical component of your army - people who carried rations/gear/etc) is usually what is seen, though it could be more or less depending on the above factors. The vassals themselves were responsible for recruiting, training, housing, and gearing up their men. The vassal also commands them - though there are of course grander tactics and strategy in play, ultimately the vassal is the one who issues orders to those he has brought with them. That being said, you still had troops reorganised into larger components, which you could argue is a 'unit' of some kind.
As for what types of soldiers you saw during the era - it is important to note that the 'Sengoku' period goes for about 130 years, and during the time you have notable changes in both strategy and pure equipment. Most notably, contact with the West introduced early matchlocks that many of those in the southwest of Japan used, while those in the northeast 'lagged' behind in technological development. In any case, you would find yourself most commmonly seeing spear groups (yari most common), archer groups (yumi most common), often gunners (teppo), as well as their mounted complements. There were cavalry mixed in with heavy infantry, and of course you can't forget those who were brought in with standards (flags) to communicate throughout the fight, as well as those involved with logistics.
Book recommendations - I'd say War in the Early Modern World edited by Jeremy Black has a good overview chapter on warfare during this time, though it does go beyond the Sengoku era it would probably help inform exactly the tactical composition of armies during this time. Thomas Conlan's Weapons and Fighting Techniques of the Samurai Warrior does have (from memory) what would be called 'battle formations', though there are errors in there (still better than any of Turnbull's works though). More recently is Fusao Nishimata's Army of Sengoku daimyo seen from modern military science, but this is in Japanese and as far as I'm aware there exists no English translation. Similarly Nakanishi Ritta History of Japanese Armor has illustrations, though I do not have a physical copy of this myself, when I read through it previously it also looked very good at illustrating how armies could be decomposed.