Do all intransitive verbs take a preposition when they used with an object? For example " She smiled at me". Smile - Intransitive verb and me - object.
As one of the comments makes clear, this question is about the definitions of the words used to describe language: most native speakers are unaware of them. Here are descriptions that I find helpful.
A transitive verb has both an active voice and a passive voice. "Hit" is an example.
An intransitive verb has no passive voice. "Smile" is an example.
Some verbs have a transitive meaning and an intransitive meaning. "Smell" is an example.
It is unfortunate that descriptions of English seldom if ever make use of the term "separable verb," a term from German grammar. By this term, I mean a verb plus a preposition that act together as a verb. "Smile at" is an example; the two words together act like a transitive verb with a single meaning.
The object of a preposition is a noun or pronoun preceded by the preposition.
The object of a verb is a noun or pronoun preceded by a verb rather than a preposition.
According to these definitions, a noun or pronoun is either a subject, the object of a verb, or the object of a preposition. But these definitions leave open the term "indirect object."
Transitive verbs in the active voice must have an object (in rare cases this may be implicit). Transitive verbs in the passive voice do not have an object.
With the exception of copulative verbs such as "be" and "become," intransitive verbs never have an object as I have defined it, and copulative verbs need not have an object as I have defined it. (Many would say that even copulative verbs never have an object; the old term is that such verbs have a copula rather than an object. This is an important distinction in languages like Latin because it affects the case of the noun or pronoun, but it is of trivial importance in English.)
So, a transitive verb in the active voice will have an object and the clause containg that transitive verb may also include prepositional phrases that contain objects of prepositions. And, ignoring copulative verbs, an intransitive verb or a transitive verb in the passive voice will not have an object although the clause containing that verb may include prepositional phrases that contain objects of prepositions.
As I have defined the term object, "She gave me the ball" has two objects for the verb "give," namely "me" and "ball." The identical meaning can be expressed with "She gave the ball to me." In that case, using my definitions, the verb has one object, "ball," and the preposition "to" has an object, "me." In such cases where the meaning is the same whether the verb has two objects or one object, we say that the noun or pronoun that must be the object of the verb is the "direct object" of the verb, and the other noun or pronoun is the "indirect object" of the verb.
I do not claim that my definitions are perfect or universal. They are inconsistent with some uses of the term "indirect object" and with standard usage concerning copulative verbs. They have the advantage of simplicity. Transitive verbs can form passives; intransitive verbs cannot. Objects of verbs are preceded by verbs; objects of prepositions are preceded by prepositions. Transitive verbs in the active voice require an object. Prepositional phrases can be used with transitive and intransitive verbs. I do not say that that is all we know on earth, but it is pretty much all you need to know about transitive and intransitive verbs and objects of verbs and prepositions.