Learn Arabic (Modern Standard): حفلة خطوبة أختي غدا. - My sister's engagement party is tomorrow.

This is a complex Idafa (possessive chain). In an Idafa with multiple words, only the final word (أختي) takes a possessive suffix or definite article. The preceding words (حفلة and خطوبة) drop their tanween and never take ال, but they become definite because they are attached to the definite final word.

How should I pronounce the ends of حفلة and خطوبة when reading this aloud?

Because both words are part of an Idafa chain and not the final word, their ending taa marbuta is pronounced as a t rather than an ah sound. You would pronounce the phrase as haflat khutubat ukhti.

If I am pronouncing the full formal case endings, what vowels go on the first two words?

The word حفلة is the subject of the sentence, so it takes a damma. The word خطوبة is the second part of the Idafa, meaning it is in the genitive case and takes a kasra. So you would say haflatu khutubati.

Why does the word for tomorrow, غدا, end with an Alif and double fatha?

In Arabic grammar, adverbs of time and place are placed in the accusative case (mansoub). The double fatha (tanween fatah) along with the supporting Alif is the standard marker for this case.