Close range infantry beats compacted infantry cavalry (e.g. siege, a cavalry blob on a hill). 2H maces = high damage but no shields, this weak to archers. e.g. mace/short sword users
Long range infantry beats staggered cavalry and short ranged infantry (e.g. open field, especially on a hill but spread out with room to maneuver). Weak to archers. e.g. greatsword/halberd users. Weak when compacted
Mid range infantry beats short or long range infantry provided they have room to maneuver. E.g. arming sword users. Weak when compacted
Other infantry: axes break shields down fast, should be used along with archers. Spears suck.
Archers: high dps, low melee skill, low ammo, accurate at medium range. Usually carry two handers as sidearms, or daggers. Check.
Crossbows: low dps, medium melee skill, medium ammo, accurate at long range. Usually carry short ranged weapons as sidearms but not always, check.
Mounted archers: suck unless it's the Noldor. Just dismount them to make them far more useful. They get around the campaign map fast but should always be dismounted
Light cavalry: suck, die too fast
Heavy cavalry: Crush infantry and archers on the open field. Moderate casualties to long range infantry. Die fast when fixed in place on a hill, by rocks/trees, short range infantry surround and tear them apart, long range infantry get alot of swings at them as they pass, medium range infantry are screwed, archers/crossbows do well but only if the cavalry is kept away from them and fixed in place at a medium distance. They're easily crushed with short or long range infantry covered by missile units but only with the kind of micromanagement the AI is incapable of e.g. compact short range infantry on a hill or staggered long range infantry (room to swing) with crossbow or archer support behind them (depends on if you think the cavalry might reach them, crossbows survive in melee better and their ammo lasts longer, archers will do more damage faster)
I find a archer/infantry setup is more than viable with the right terrain.
Set infantry 2 lines thick halfway up a steep hill. You want the enemy cavalry to come to a dead stop when they hit them. Because the enemy cavalry will be packed side by side like sardines they cant really swing their weapons or disengage except from the flanks. Archers go on the top of the hill behind them and fire down on the fixed in places enemy cavalry. You will want your infantry to use close ranged weapons for this setup usually but not many units in PoP get close ranged weapons, only really the noldor infantry which is why they rock in melee
If you must fight them in the open field or if you are outnumbered then halberds are the way to go. The outnumbering cavalry will easily get around your flank and disrupt any archer formation so don't even bother. Halberds have a long range but also a very good minimum range. They hit hard and their jab will stop a horse in it's tracks. The problem with short ranged melee weapons vs cavalry is that unless the cavalry is fixed in place they'll never get to hit them, or only one will ever really get to swing at a time and for low damage
Long ranged melee weapons have more opportunities to strike and do more damage, especially two handed such as greatswords, greataxes, greathammers or halberds. You ideally want one with a good max and min range and good damage, because the horses like to ride right up in your guys faces where they cant swing at them, then ride off and whirl around, the halberds will have plenty of chances to swing and everytime a cavalry charge is broken each enemy cavalry will be targetted by 3-4 halberdiers who are all in range, and many strikes will hit the rider for heavy damage and with high shield damage too! This works poorly against surrounded cavalry or compacted cavalry because there's no room to swing, but extremely well when you are caught in the open and the cavalry is skirmishing back and forth through your troops. Halberds are the way to go when dealing with cavalry heavy factions like D'Shar and Jatu who tend to swarm with cavalry pretty fast. They do have horse archers though so you had better be prepared to pick off the horse archers yourself, two handers are vulnerable to missile troops
In general keep in mind, horse archers are meant as a hard counter to two handers, but currently only noldor AI can pull it off. Heavy cavalry are the hard counter to short ranged infantry caught in the open or missile units that aren't being defended (or can't be defended) by infantry. Compacted infantry on a hill will stop a cavalry charge and compact it, and can hold defensive positions well for archers to fire. Infantry with short ranged weapons massacre compacted cavalry or medium/long range infantry and are the best siege rush unit, but most mods nerf or leave them out entirely. They're usually mace or hand-pick wielders (axes are surprisingly mid ranged) but noldor short swords are close ranged too, and 2H maces are close ranged high damage but lack shield support. Two handers are the cavalry counter provided they have room to swing, and are best deployed loosely on a hill so they have room to swing and the cavalry is still slowed down a lot, they do well on the open field too. Archers vs crossbows, archers are high dps and short range and run out of ammo fast, crossbows are low dps, high range, tend not to run out of ammo, and are usually more competent in melee so don't need to be protected as much
And those are the general rules. The AI ignores them completely so tailor your army to the enemy's weaknesses. Avoid field battles it's a better strategy to chain siege with 10+4 engineering. This means close ranged infantry and crossbows are your bread and butter and this applies to all mods. Heavy cavalry are for mopping up individual lords. If you want to take out those pesky warbands then either hope your marshall engages them, or if you become marshall then wait for a siege to start, hide somewhere near it with your warband til all the enemy lords enter the siege then move over quickly and fight the defensive siege. When it is over you will all be outside in a cluster so engage the nearest lord. You'll be in command of all your allies troops so remember the strengths and weaknesses and deploy accordingly. Try to protect your own units and bring units that are a strong counter to the enemy