My wife and I will be in County Kerry and County Galway/Mayo in May. I'm working on distinguishing between the Munster and Connacht dialects. Are both of these sentences telling age correct for Munster? Thanks for your time!

1. Táim trí bliana agus tríocha d'aois.

2. Táim tríocha trí bliana d'aois.

Not sure about munster dialect, but in general:
-- In the first one, you don't need the "agus".
-- The second one is not good Irish.

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

No, you definitely need agus (or shorter is): trí bliana is tríocha
or you could say: trí bliana tríochad
or more traditionally: trí bliana déag is fiche, trí bliana déag ar fhichid

I've never heard anyone use "agus" there, but yes of course either "'s" or "is" is needed. I wasn't thinking clearly, just reacting to the "agus".

_________________
I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.