I recently began learning from Mounce's Basics of Biblical Greek. He says you should learn the lexical form, stem and article of a noun. Can these things not be worked out if you just learn the lexical form?
I want to start memorising the vocabulary as I mean to go on, so my question is which forms of the noun do I have to learn to be able to read and translate from Koine Greek?
As Mounce said, it's important to learn the articles of all the nouns because it's not always obvious what they should be from just the lexical form. A good example of that is ἡ ἔρημος (the desert, wilderness) which is a second declension noun. Most second declension nouns are masculine and have ὁ for the article, but ἡ ἔρημος is feminine and takes the feminine article.
It's also important to learn the stem of all the nouns because the lexical form isn't always a good indication of what it should be. For example, ἡ νύξ (the night) has for its stem νύκτ-, which isn't obvious. Another example is ὁ πατήρ, which might lead you to think that the stem would be πατ-, when actually it's πατρ-.
One good way to learn the stem is to learn the genitive singular and drop the ending. For example, the genitive singular of πατήρ is πατρός. By dropping the ending -ός, you get the stem πατρ-.
The stem can't always be derived from the lexical form. For example, in the third declension, the nominative singular (i.e. the lexical form) is usually formed by adding an -s, which often obscures the end of the stem:
And, unfortunately, the lexical form can't always be derived from the stem either. This often goes back to something called Proto-Indo-European ablaut, and doesn't follow any regular pattern in Greek; it just has to be memorized.
So you need to memorize both. Similarly, the gender isn't always predictable from the form—there's no good way to know which of the nouns I just listed are masculine vs feminine without memorizing each one—so you'll need to learn that too.
