am • is • are // was • were
BE is used in progressive/continuous tenses and for passive voice. When a continuous tense becomes perfect-continuous, BE is manipulated even though it’s no longer the first aux verb. BE is also used for the future be going to.
BE can also be a main verb. It is a linking verb, which means it connects the subject to another noun or to an adjective. BE is unique because even when it serves as the main verb of a sentence, it can be manipulated the same way as an auxiliary verb. This is done only when no auxiliary verbs are present (i.e., when BE is the main and only verb of a present simple or past simple sentence).
do • does // did
DO is used when no other auxiliaries are present. It’s often used in questions and negative sentences in the present simple and past simple tenses. Whenever you add another auxiliary verb (like HAVE for perfect tenses), DO disappears.
DO can also be a main verb meaning to perform, execute, initiate, etc.. Unlike BE, it cannot be treated as an auxiliary while serving as a main verb. If no auxiliary verbs are present, you’ll need to add another DO specifically to fill that role, which means you might have two DOs (one as an auxiliary verb; one as the main verb).
have • has // had
HAVE is used in perfect tenses. It also comes after modals to make them past. When HAVE is added for either of these reasons, the verb that immediately follows should be in the perfect participle form.
HAVE can also be a main verb, in which it denotes possession. In some UK dialects and in more archaic usage, it can be manipulated as an auxiliary verb while acting as the main verb, just like BE can. However, if any true auxiliaries are present, the first of those should be manipulated instead.
can • could // may • might // must
will • would // shall • should
Modals are used to express possibilities, expectations, permissions, obligations, abilities, and tendencies, as opposed to stating what is. Future tenses typically use modals verbs.
If modals are present in a sentence, the (first) modal is manipulated when creating a question (they take priority over BE, DO, and HAVE). For multi-word modals and semi-modals (e.g.: must have, be able to, had better), only manipulate the 1st word of the modal phrase.
Auxiliary verbs are the key to many structures; often the word that needs to change when altering the structure of a sentence is the auxiliary verb (or the 1st auxiliary verb, if there are multiple).