Learn Finnish: Hyvää iltaa, nainen ja mies! - Good evening, woman and man!

They are in the partitive case. In Finnish, common greetings like 'Good evening' or 'Good morning' use this case because they are actually shortened versions of a longer thought, like 'I am wishing you a good evening.'

How do I know whether to add an 'a' or an 'ä' to make these partitive forms?

This is due to a rule called vowel harmony. The word hyvä contains the vowels 'y' and 'ä', which are front vowels, so it takes the front-vowel ending -ä. The word ilta contains an 'a', which is a back vowel, so it takes the back-vowel ending -a.

If the greeting uses the partitive case, why are nainen and mies just in their basic dictionary forms?

The words nainen (woman) and mies (man) are the people you are addressing directly. In Finnish, when you call out to someone or address them by their name or title, you always use the basic, unchanged form (called the nominative case).