My family (cousins, uncles, aunts, etc.) and I are looking to get a tattoo with the word "family" in Irish Gaelic. I have seen the words muintir and teaghlach in the search that I have done. Please help with the correct word to use! Also, if it is muintir, do we spell it as mhuintir?

Not all speakers agree precisely on this, but in Irish, teaghlach usually refers to the folks in your household; muintir usually refers to a broader category of people (depending on the context, it can mean your whole family or clan, beyond just the ones in your home, or even your community or your “people”); and there's a third word, clann, which also can mean different things depending on the context. In everyday speech it usually now refers just to your children and/or descendants, but in certain contexts it can refer to an extended family or "clan" ( a word which came into English from either Irish or Scottish Gaelic).

The word "mhuintir" can't stand by itself. The "h" would be added only in certain contexts, such as after the words "mo" ("my") or "do" ("your"), giving you mo mhuintir ("my family") or do mhuintir ("your family"). It's there to mark a sound change which occurs in those cases (in those particular cases, the "m" sound becomes essentially a "w" sound).

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

I'd think, muintir (family) or mo mhuintir (my family) is the best choice for your core familiy.

That would be my choice as well. I lean a bit toward "mo mhuintir" (my family) or "ár muintir" (our family) to make it clear that you're not using it in a more general sense.

Redwolf