Hello Vanjasper and a warm welcome.
I am not trying to be clever or patronising here.

I cannot help at all with Irish records, as I have never come across any of my family from there.
The very best place to start is at the beginning- with your birth certificate. From there your parents marriage certificate should give you your grandfathers names. Your parents birth certificates should give you all of your grandparents names.
Treat all the family stories with care, as they do get embroidered over time.
I would be looking for grandad's army records.
ancestry.co.uk/ has army records, but it is a pay to view site.
I will warn you in advance though----this hobby can become very addictive, and sometimes expensive.
If you want to post up Grandad 's name I am sure many here will help with your hunt.
Best of luck.