I'm not too sure what your asking for beyond medieval army compositions but it looks fun so I'll give it a try.
From my perspective an army should consist of 1-2 lines of shield men followed by people with either pole arms or spears. The reason for this? The shields protect your people from the front and maybe some arrows, the pole arms are able to reach above and over the shield or through small gaps in the shield and stop people from getting close. The best example of how terrifying this can be would be in the Game of Thrones scene where John Snow and the Wildlings are being surrounded by this composition. It is extremely hard to break if you value your life, because the person who runs into a shield will more or less likely die.
Now what you want after this is your standard soldiers, probably using spears over swords. Why a spear over a sword? Its easy to train someone to use a spear, it requires less metal (only the tip) and it has a very long reach. In an army vs army situation, you point your stick at the enemy and the enemies are going to push themselves onto you. Thats not to say you won't have a sword, a sword would probably be your secondary weapon, so once the spear is used, you pull it out and start to fight it out.
After your shield wall, your main body of soldiers, you will mostly likely have your cavalry or archers. The cavalry would of course move out and try attack the flanks of your enemy. There won't be much shields on the sides because the enemy isn't expecting an attack from there. Of course, your cavalry would have a hard time taking on a row of spears so they will probably have a diminished role and be attempting to attack the back lines where your defenseless archers are situated.
Your archers are situated in the back, shooting away. You will of course have to be able to estimate their approximate range and make sure that your shield wall isn't going to be close to it. You don't want a stray arrow hitting your allies, so your archers are going to be aiming into the main body of the enemy, far behind the shield wall.
So thats roughly what I could consider as a basic formation. Of course this changes with terrain and resources available and then you also have to develop army formations based on who they are fighting. Going back to Game of Thrones, in the scene when Stannis defeats Mance, he easy wins because his armored knights are able to easily slaughter their way through barbarians who don't have the teamwork and training to deal with armed cavalry (e.g stick together in clumps and strike out at once and spears).
Then after you have terrain and resources taken into account you want to adjust your army formation more. Should they be in one big clump, maybe spread into smaller clumps. How would different units support each other and what are their different objectives.
Back to your suggestion of pikes and crossbows along with light cavalry, lets analyse how it plays out. My pikes are good vs the enemy cavalry but they can't move forward easily because they can either carry a shield and pike which means if they are hit by the horses when they are moving they aren't going to be as prepared or they can use just a pike which leaves them vulnerable to the enemy crossbows. If the pikes move out, your crossbows are suddenly vulnerable to the Calvary who are faster and once they are on top of your crossbow men it will be a slaughter. Your crossbow men can only sit there and wait, because if they move into the range of the enemy crossbowmen they will get shot first and if they move away from the pikemen they will be attacked by cavalry, the winner would end up being the defender.
If instead one army had a shield wall with polearms/spears/pikes mixed in, they can suddenly advance forward and start to put pressure on the defender. The shield stops the bolts with the polearms people hiding behind the tall shield. The cavalry are repelled when attacking the shield due to crossbow support and the long reaching pole arms. Once the shield wall is within pike range, the enemy cavalry and shield wall charge in from the front and sides. Behind this the enemy crossbow men advance and rain down bolts on you.
What I'm saying is that your army lacks shields, a key item to defend yourself from ranged and close quarter combat. It doesn't matter if its a Captain America style shield of if its taller then the person carrying it, it gives your men a safer way to move forward.
Also remember that pikes are good vs Knights on horses and in shiny armor. They would be less effective against infantry because they aren't designed to deal with infantry. A spear is shorted and more thurstable which makes infantry fighting easier but fighting people on horse back harder.