I'm going to stick to Ireland for this answer.

Modern humans arrived in Ireland after end of the last ice age, which was about 7000 BCE for that region, the Mesolithic (there is some very limited evidence for an earlier date, but this isn't the place for that discussion). They came from Britain, were serviceable seafarers, and were sea-oriented as far as their subsistence. The Irish Sea had already re-isolated Ireland as an isle but was much more shallow then than it is now. They were hunter-gatherers and ranged all over the island, and we have archaeological evidence of them from County Cork in the south to Donegal in the north. Their population was very, very small, probably less than ten thousand, and Ireland was not a very pleasant place then, with harsh climate and a general lack of game. In addition to ocean production, they gathered nuts and fruits as best as they could and hunted wild boar in limited amounts with simple microlith blades/points and flaked stone tips.

By about 4500 BCE, the Neolithic "suite" of features had arrived in Ireland, either as ideas or as new migrations of peoples (or both): limited cultivation of crops, especially cereals; herding and population augmentation of animals brought from Britain and Scotland, especially cattle, sheep, and goats (and the introduction of such animals as red deer); population increase, perhaps into the low hundreds of thousands; house-building and clustering (though pastoralism remained largely mobile throughout the Irish Neolithic, as well); and megalithic construction. Céide Fields in County Mayo is worth special mention: it is maybe the most extensive, oldest, and best-preserved example of Neolithic farming layout anywhere in the world.

In this way, the Irish Neolithic was similar in most aspects to its cousins in Europe, like the La Hoguette Culture of NW France: some combination of "Neolithic Revolution" factors, including agriculture, herding, increased social organization, advances in pottery and other technology, megalithic building, etc. The introduction and development of these aspects happened gradually, over the course of several thousand years, and varied sharply by region, as is the case for the Neolithic the world over.

Also in this period were built the famous megalithic structures of Neolithic Ireland. Most have been interpreted as religious structures and/or tombs. There are court cairns, with open-air central courtyards (they look a lot like the tombs in Skyrim), portal tombs (which look like a henge now, but were originally covered with earth to form mounds), etc. These structures situate the Irish Neolithic with other megalithic builders (like those on Sardinia, Malta, etc). Over a thousand of these survive in some form or another, dotted all over Ireland. As for written records: there are none for Ireland until the end of Antiquity, 5th century CE. Until then, it is all archaeology.

This brings us to the Early Bronze Age, about 2400 BCE in Ireland. New people arrived, bringing with them the "Bell Beaker" culture and new ideas. They replaced the Neolithic peoples in Ireland, as happened similarly throughout Europe. New arguments have suggested that these new-comers were Indo-Europeans, bringing with them the Proto-Celtic language. It used to be thought that Proto-Celtic did not arrive until the Iron Age, about 500 BCE. Either way, about 2400 marks the end of the Neolithic peoples of Ireland and the beginnings of the people/culture which would become the "Celts" encountered by the Romans in the 1st century CE.

A good and recent(ish) generalist source for the Prehistory of Ireland is: O'Kelly, Early Ireland : an introduction to Irish prehistory (Cambridge 1989).