Hey everyone I'm new to this forum. But my fiance and I are going to get matching tattoos and being that both of us have Irish heritage we loved the idea of writing something intimate on our bodies in the traditional Irish gealige.

We found this on multiple sites:

A chéadsearc (uh KHAYD-shark): First love (not “first” as in a series, but “first” as in “primary”).

And we have also found it in the possessive form "Mo chéadsearc". We wanted to verify the meaning with people who know what they're talking about and also ask what would be the better or more proper way to put it.

Also side note she does know someone who has family in Ireland and knows how to speak the language and they told us they couldn't find the word in an Irish dictionary. I'm not sure if that's something we should be concerned about or not. Please help!

Thank you all, Ty and Becky.

It depends.

a chéadsearc means "(my) first love" in addressing the person
mo chéadsearc means "my first love" everywhere else, except in addressing.

First love is of course céadsearc (https://www.teanglann.ie/ga/fgb/céadsearc)
the first love: an chéadsearc

This term is in all the dictionaries that I have looked at. That said, there are other ways to express the feeling as in "An bhean is annsa
liom féin" > The woman I love best / "An fear is annsa liom féin" > The man I love best (from Dinneen's Irish English Dictionary). You can Dinneen's https://celt.ucc.ie/Dinneen1sted.html and plug "love" in the search function to find all sorts of usages and phrases. Of course, check back here for confirmation if you choose to do that.

Some people, when looking words up in an Irish dictionary, forget to take out the "h" which is caused by "lenition" of the initial consonant in various situations, such as after words like "a" and "mo" in the examples you have.

céadsearc - nominative case [form used in dictionary]
a chéadsearc - lenited form [based on the particular context in which the word is used]

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