Ireland was an undesireable posting and yet it became the biggest drain on manpower for the English army at the time. At its peak, it would have had 18,000 soldiers whereas the campaigns in the Low Countries never saw more than 12,000 English soldiers deployed.

The army in Ireland was reinforced by a mixture of stripping garrisons along the Scottish border, sending troops from the Continent (withdrawing them from Brittany or the Low Countries), and local raisings in counties in England. The latter were very unpopular and commanders would find some of their new command would have disappeared before they even reached the docks to set sail. Companies were always understrength as it could be very difficult to source replacements. Captains were also notorious for "dead pays", claiming they had 100 soldiers so they could draw wages for 100 when in reality there might only be 90 in the company.

Sheriffs in counties would be given a quota and they'd go around grabbing the unemployed and undesireables. Criminals were also used, it provided a nice chance for a county to "clean up" and meet their obligations. This caused no end of trouble in Ireland as the officials there are recorded as frequently complaining about the quality and quantity of troops they were provided with.

There was also use made of Irish troops. These would either be Irish troops that were armed and equipped by the English crown or auxiliaries from friendly chiefs who fought in the more traditional manner. One problem with both the former and the latter was that the English could never be quite sure of trusting them. Local recruits had a tendency to disappear with their weapon if they felt like it, either going over to the other side or becoming a bandit. Gaelic troops also had a reputation for being flighty as their chosen method of warfare at the time was predominantly skirmishing.

Officially, from the 1570s on, there were meant to be no more than five or six Irishmen in each company of crown troops but in practice this was rarely followed. Irish soldiers were cheaper and readily available, it was a great opportunity for a captain to bolster his forces and make a bit more money. Several companies would have ceased to exist if this policy had been followed to the letter.

Both galloglas (heavy infantry with axes/swords) and kern (light infantry with a bow/javelins/spear) were used by the English crown as they were easily sourced and raised, they were everywhere in Ireland. By 1600 most of these had adopted the pike or musket as their weapon, one of Tyrone's achievements.

Pikemen started off in 1500s Ireland quite heavily armoured but seem to have ditched more and more of their armour as the century went on. There's frequent mention of pikemen in Ireland forgoing most of their armour in favour of mobility and speed. Pitched battles weren't the norm. Raids, skirmishes, sieges etc were far more common so soldiers wanted to be able to move fast like their Irish opponents. Sketches of pikemen from the late 1500s show them only with helmets and breastplates with no armour from the waist down.

Musketeers and calivermen were similar. Helmets were popular but some wouldn't wear much more in the way of armour. Speed was essential, campaigning in Ireland involved long marches over rough terrain. The caliver was also preferred over the (then) bulkier musket. Mountjoy requested 1000 additional calivers from London in 1600 as opposed to 100 muskets. The caliver was much more suited for Irish warfare and both sides preferred it to the cumbersome muskets.

Sources:

Elizabeth's Irish Wars - Cyril Falls

Irish Battles - Hayes McCoy

The Nine Years War - James O'Neill