In ainm an athar agus an mhic agus an spioraid naoimh

That's how I was taught this prayer, but shouldn't it be an t-athar agus an mic? I'm guessing that it's an old fashioned form. Anyone have any information on this?

No your first sentence is right, it is "an athar" and "an mhic" and "an spioraid naoimh" : they are in the genitive case (the name of...).

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

To express what Lughaidh has given you another way:

an t-athair is "the father"
an athar is "of the father"

an mac is "the son"
an mhic is "of the son"

an spiorad is "the spirit"
an spioraid is "of the spirit"

Those are for any old dad, son and spirit. As formal titles:

an tAthair is "the Father"
an Athar is "of the Father"

an Mac is "the Son"
an Mhic is "of the Son"

an Spiorad Naomh is "the Holy Spirit"
an Spioraid Naoimh is "of the Holy Spirit"

(Note how the capital A affects the use of the hyphen.)

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Go raibh maith agaibh