I am attempting to answer this myself, would update the question and answer as I figure out.

à or de after a verb

Basically à denotes that the verb is applied to the object of the sentence, hence we have examples like

• repondre à (reply to)
• partir à (leave for)
• aller à (go to)

On the other hand, de usually means the object being the source of some verb, for instance

• rêver de (dream of)
• parler de (talks about)
• partir de (leave from)
• venir de (come from)

(FIXME I have not figured out a reliable way to tell if a/de is needed, and which to use for now, to be updated, hence this question. Verbs that describe movement like partir is easy, but not for the rest)

nouns after de

if de follows a verb (type 1)

In this case, the following noun is usually singular, notice the noun is in plural form when translated to English.

• changer de travail/carrière (changing jobs/careers)

FIXME why? because these are abstract/conceptual?

if de follows a verb (type 2)

sometimes le/la/les is needed

• aller *a la* pharmacie
• aller *au* supermarché
• aller *aux* toilettes
• venir *du* Japon
• venir *de la* France
• venir *des* États-Unis

in these cases, it is also possible to construct similar sentences, replacing the $something into forms like

• aller a son bureau
• venir de ton bureau

FIXME why? because they are concrete objects?

if de follows a negated verb ne _ pas

In this case, the following noun is usually singular, and it is also singular when translated to English

• n'avoir pas de machine (does not have a coffee machine)

(FIXME elaborate more?)