Looking to see the translation of "our little family" and " our family"

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 tribe or your “people”). The word clann can also mean family (that's where we got the English word clan from), but in modern Irish it is generally used only to refer to one's children or the descendants of someone.

Assuming that you mean your household, when you put the word "our" (ár) in front of teaghlach, it causes a sound change at the beginning of the word, which is indicated in spelling with a "d", so you'd have:

ár dteaghlach
our family

So, for the full expression you want, a literal translation would be:

ár dteaghlach beag [most adjectives follow the noun in Irish]
our little family

There may be a better way to say it than just literally, though, so wait to see what comments others may have.

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