How does one go about applying for residency and/or citizenship in Ireland?
Answer:
It depends if you are not from an EU country and don't have parent or grandparent from Ireland, the chances of legal immigration to Ireland are very slim.
You can try to find a job there and have the employer apply for a visa, residency permit. Unless you are in a very sought after field, this is an unlikely avenue. Companies have to demonstrate that no equally/more suitable candidate could be found in the EU.
You can marry an Irishman! Marriage would give you an almost automatic right of residency.