[subject] + 既 + [adjective] + 又 + [adjective]
[subject] + 既 + [verb phrase],也 + [verb phrase]
[subject] + 既不 + [verb phrase],也不 + [verb phrase]
既…又 and 既…也 say that one subject has two qualities or does two things: "both … and". 既 goes before the first, and 又 or 也 before the second. Both halves share the subject, which comes before 既.
With 又 the two halves are equal partners, often two adjectives packed close together. With 也 the second half is added on top of the first, and it is the usual choice when the halves are longer phrases or a recommendation: "you need X, and you also need Y".
With 不 in both halves the frame means "neither … nor".
既 is an adverb, so it comes after the subject and can't take a subject of its own in each half. Two adjectives or verbs are joined with this frame, not with 和.
又…又 means much the same and is more casual. 既然 "since" is a different word that happens to start with 既.