Can anyone tell me why these change and why there is an h added and t's etc
Or have a referance guide? TIA

The short answer is that it's just the grammar of the language. The initial sound of a word can change based on the type of word that comes before it. These changes are known as initial mutations, and they're a unique feature of the modern Celtic languages. There are a lot of different types of words which can cause the following word to change its initial sound, so it's generally better to memorise the examples you're learning from rather than to try and digest the grammar, at least at an early stage of learning.

To give you a few examples of how this works, based around the words you've listed in the title of this post:

Athair = "father", but an t-athair = "the father", a hathair "her father", and ár nathair "our father".
Máthair = "mother", but an mháthair = "the mother".
Mac = "son" but a mhac = "his son".

If you look up "initial mutations" in Irish you can find more detailed information about this phenomenon. These can be further divided into initial consonant mutations, and vowel mutations. The initial consonant mutations are known as lenition and eclipsis, while the vowel mutations are known as t-prothesis and h-prothesis. The insertion of the "h" in examples above like an mháthair and a mhac is lenition. Eclipsis is the insertion of letters before the initial letter of a word, like in ár nathair. Finally, you have an example of t-prothesis where a "t" is inserted before the vowel in an t-athair, and similarly of h-prothesis in a hathair.

Go raibh maith agat, slan go foill!

máthair is a feminine noun, so you have "an mháthair". Mac is a masculine noun, so you have "an mac". So lenition does depend on the gender of the noun. But it's easiest to learn the noun with the article: an mháthair, an mac.

athair is a masculine noun, so you have "an-t-athair" where the noun starts with a vowel. This is "t-prefixation".
amharclann (theatre) is a feminine noun, so you have "an amharclann" with no t-prefixation.

This is the briefest overview of some basic rules. I've only covered the definite articles, but as Ade shows there are other rules, including with the possessives "his, her, its, theirs" and many more besides.

My honest advice is simple: do not try to learn Irish using only Duolingo -- it just doesn't work.
This is just one of many grammar points that Duolingo gives absolutely no information on, and expects learners to do the impossible and intuit it from context and examples. I'm currently using Duolingo for Welsh, and even though I speak Scottish Gaelic pretty well, understand a reasonable bit of Irish and have even studied Welsh before at university, there's lots of times where I'm just completely lost in the dark about grammar.

Duolingo was set up by a team of computer scientists, not language teachers.

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A language belongs to its native speakers, and when you speak it, you are a guest in their homes.
If you are not a good guest, you have no right to complain about receiving poor hospitality.

ár nathair = our snake
ár n-athair = our father

Funnily enough people say Athair Nimhe for snake as well.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

I agree about Duolingo. Just because it is there does not mean that it is the best or even a good way to learn a language like Irish.

I used Duolingo for a couple years to learn Irish with very little success. A couple months ago I gave up Duolingo and started working through Progress In Irish. I am making far better progress. To be fair Máiréad Ní Ghráda's course isn't designed for self-learners and I have lost count of the number of times I wished I had a teacher to ask about a grammatical point. I have to spend quite a lot of time looking stuff up on online grammar guides (nualeargais.ie, for example, through wayback machine now unfortunately), online dictionaries (teanglann.ie, for example) and pronunciation guides like abair.tcd.ie/ga/synthesis and fuaimeanna.ie. I use forvo.com a bit but as it doesn't tell you where in Ireland the speakers come from it is less useful to a learner like me. I combine this with using Anki a free flashcard program for recall testing and spaced repetition.

Well Bríd Eilís on Forvo is a native speaker of Connemara Irish.