One of the main differences between Irish and English is article usage, and it is a source of errors and headache for all learners. I didn’t learn it overnight either, so try not to lose heart. It is possible to learn it.
To start with, remember that while we say in English, for instance, “the president of Ireland”, “the managing director of the company”, in Irish genitive constructions, one definite article is enough. So, in Irish we say:
the president of Ireland = Uachtarán na hÉireann
the managing director of the company = stiúrthóir bainisteoireachta an ghnólachta
To add an extra article (*an tUachtarán na hÉireann, *an stiúrthóir bainisteoireachta an ghnólachta) is out and out wrong, and it is wrong in all the Celtic languages. It is only allowed when the first noun is qualified by seo, sin, or úd – i.e. you can say “an tUachtarán seo na hÉireann“, but it is not required – the first article can be omitted: “Uachtarán seo na hÉireann” ‘this president of Ireland’.
Note that if a noun followed by a definite genitive attribute is indeclinable. If it is put in genitive position, that genitive will only be expressed by leniting the first consonant, if it can be lenited. Compare:
Aerfhórsa na Stát Aontaithe “the air force of the United States”
Aerfhórsa Stáit Aontaithe Mheiriceá
“the air force of the United States of America”
Note that as a proper name Meiriceá is inherently definite.
A noun should not be modified by two definite genitives. This is why I find the recommended form “Duais Nobel na Síochána” (“Nobel Peace Prize”) objectionable: “Nobel” is a proper name, which means that it is a definite noun, and after “duais”, it is a genitive. I would prefer “Duais Síochána Nobel”.