I know this is an easy one. I am getting a tattoo in honor of my father and grandfather (both Irish) I've serched the internet for the word "father" and all I could find was: athair. Is that correct? I would like to have the correct word with fadas and everything. :)

"Athair" is correct, no fadas. It is derived from an Indo-European word similar to Latin "pater". The 't', lying between two vowels, was softened to 'th', as has happened in many Irish words. For that matter, it has been softened to 'th' in English. (The change from 'p' to 'f' has happened a lot in Germanic words.) In French, the 't' is lost altogether.

And the letter 'p' has been lost from many Irish words. In some words, 'c' replaced 'p'.

In Celtic, the Indo-European p has been lost in all words.
Irish c's come from Indo-European Kw (which has become P in P-Celtic languages and Q in Q-Celtic languages, as Irish).

When you have p's in Modern Irish, anyway they don't derive from Indo-European p's, but from other stuff (loanwords, phonetic evolution of other sounds, etc).

as an aside point, th can be h medially in hiberno-English, maybe following on from Irish, example: method 'mehod', though I think only in 2 syllabic words were there is a short vowel next to th

In Belfast: nothing > nahin, I think.