"The first girl's dress is the prettiest."

Translation: Rochia primei fete e cea mai frumoasă.

[JamesTWils]

I had thought, simply from the examples given, that the genitive definite of fată would be fetei. Does anyone have a link to a simple paradigm for Romanian nouns, so one can get a sense of the declension, particularly of these definite feminines?

[potestasity]

It is, you are right. However, when placing an adjective/ordinal numeral before the noun, the article is applied to that, instead of to the noun. Thus, you have "primei" instead of "prime" and "fete" instead of "fetei".
https://dexonline.ro/definitie/fat%C4%83/paradigma - this might be useful to you. You can search any word and click on "conjugări/declinări" to see declensions or conjugations.

[JamesTWils]

Wow, that is very interesting in a Romance language. Thank you very much for the link.