What are the visa requirements for a UK citizen wanting to stay in Ireland for longer than 3 months, for non-work, non-business reasons?
There are no visa requirements for British citizens wishing to stay longer that 3 months in the Republic of Ireland.
Common Travel Area (CTA)
Under the CTA, British and Irish citizens can move freely and reside in either jurisdiction and enjoy associated rights and privileges, including the right to work, study and vote in certain elections, as well as to access social welfare benefits and health services.
Except for voting for the President of Ireland (because Irish citizens can't vote for the UK head of state), a British citizen has all the rights as an Irish citizen while they are in Ireland and vice versa.
The primary difference in moving from Manchester to say, Liverpool, and moving from Manchester to Dublin is that you may have to open a new bank account rather than just change the address and also register with health/tax/social security authorities.
As you want to travel for non-work reasons, you just go to Ireland and do the things you went there for. Being a British citizen, you can live in Ireland forever without the need to get any residence permit.