"You go to the airport."
Translation:Téann tú chuig an aerfort.
After the simple preposition chuig you use the dative case, so there is no mutation after an - chuig an aerfort. After the phrase go dtí you use the nominative case, and in the nominative case, mascuine nouns that start with a vowel get a t-prefix after an - go dtí an t-aerfort.
"You go to the airport" can be translated as either as Téann tú chuig an aerfort or Téann tú go dtí an t-aerfort.
So, there are two (probably more) types of ways to go at sentences like this. One uses a 'phrase' ('particle' + 'adverb') and the the other uses a preposition.
In their following nouns, 'Phrases' require the nominative form and prepositions require the dative form. Do I understand that correctly? Is that a fairly reliable rule?
You're making a "rule" for something that probably only applies to go dtí an and chuig an.
At least I can't think of any ('particle' + 'adverb') phrases that would be the equivalent of as or ag or leis, etc.
The only "rule" you need to remember is that simple prepositions + an cause eclipsis. Go dtí isn't a simple preposition.
I wanted to write 'An t-aerfort' as well, and was slightly proud of myself for recognizing it as a feminine noun (still not sure how I did it), but it was not given as the desired solution. I figured I had just jumped the gun and got it wrong, as I'd never encountered aerfort after a definite particle before, but does this mean I actually got it correct? If so, why didn't duolingo give that as the answer?
Téann tú chuig an aerfort or
Téann tú go dtí an t-aerfort.
As pointed out in the previous comments, after prepositions like ag, ar, leis, chuig etc and the singular definite article an, you get eclipsis for words that start with a consonant and no change for words that start with a vowel, regardless of the words gender (except for some words that start with s).
You follow the normal rules for the nominative case after go dtí.