Culturally, the answer to your question is convoluted and deeply personal.
Legally with regards to nationality, it's clear cut, albeit complicated to explain. And because I used to work in this area and am somewhat rusty at explaining...
Your grandparents were born in Ireland - they are Irish.
Your father was born in the UK to Irish parents. He has dual nationality. If he was born here prior to 01/01/1983, he is British purely because he was born on British soil (jus soli entitlement ended on 01/01/1983 with the British Nationality Act 1981 coming into force).
If he was born after 01/01/1983 he is British because he was born in the UK to settled parents ('settled' being reference to a specific immigration status - indefinite leave to remain - which Irish nationals hold as soon as they move here). Doesn't matter which way around, your father is British.
He is also Irish by descent (again, 'by descent' being a specific immigration term with defined meaning within UK and Irish immigration laws) because he was born outside the island of Ireland, to parents who were born on the island of Ireland.
He doesn't have to do anything to claim his Irish nationality, he was born with it. He can apply directly to the Irish passport service and would be given a passport.
You, are not Irish by nationality. The automatic transition stopped with you because you were born outside the island of Ireland to a parent who was also born outside the island of Ireland.
You can, however, apply to go on Ireland's Register of Foreign Births and from there, apply for Irish citizenship and your Irish passport. The 'Irishness' then stops with you if you have children already. They can't get Irish citizenship through you. If you apply to go on the Register before you have children, they can then apply in turn to go on it and the entitlement to Irish nationality is preserved.
Clear as mud, innit. 
By the way, I am exactly the same as you and have an Irish grandparent born in Ireland and an Irish by descent parent born in the UK. I think of myself as British with Irish heritage.