To my knowledge 'Sé' is for the subject and 'é' is for direct object in most cases. So 'Taitnionn sé é' would be 'He shines it'. You can sometimes tell if a word is masculine or feminine by the ending. Most nouns are masculine but most often nouns ending in a slender consonant, -eog, -óg, -lann, multi-syllable nouns ending in -acht or -íocht, and country and language names are feminine.
I think you use é with nouns or adjectives, and the word is (remember "is cailín í?"). If you were saying it is shining, maybe you'd use é, but taitníonn is a verb. That's my guess, based on what we've learned at least!
As for gender of a word, it seems to have to do with declension. I've gotten a bit frustrated with it myself, so I found this link, and maybe it'll help. http://nualeargais.ie/gnag/subst3.htm
Rachel, just like French and German, every noun in Irish is either masculine or feminine. That's why bean becomes an bhean after the singular definite article (feminine nouns are lenited after an in the nominative case), but fear becomes an fear (masculine nouns are not lenited after an in the nominative case).
Look up any noun in a good Irish dictionary (such as the Foclóir Gaeilge-Béarla), and the gender of the noun will be indicated at the start of the entry.
Since when? Sorry but i've been learning irish at school for 11 years, and never once has gender of words been mentioned. I thought it was weird when in secondary school we started learning french and german and they had genderised words. I just assumed irish didnt. Jeez, like not my fault I've been learning something 11 years and no teacher ever bothered to tell me this information.
I had exactly the same reaction when I encountered this idea of gendered nouns when I started using Duolingo, and I've checked with lots of other people who learned Irish in school, and many of them were also surprised. So there's nothing unusual in your experience, but the gender of nouns really does matter - it's why we say an bhean bhocht rather than an bean bocht, and why we say muintir na hÉireann rather than muintir an Éireann.
Taitníonn é is something like "him shines" - it is not grammatically correct.
Irish does not differentiate between "he" and "it" - sé can mean "he" or "it" depending on context, and é can mean "he", "him" or "it", depending on context.
That doesn't mean that you can choose between sé and é whenever you want. é can't come immediately after a verb - you have to use sé for the subject of a verb, whether the subject is "he" or "it".