The easiest and most common way is the direct object. In this case there is no preposition at all: John sees the tree. For unknown reasons we call that a case, in grammar books it is called an accusative, although there is not a big difference between "John see the tree" and "John thinks of her". In both cases we connect a complement to a verb. In the first case we do it without a preposition and in the second case wie do it with a preposition. (For those who speak Spanish: A personal accusative, an accusative that is marked with the the preposition a exists in Portuguese only in combination with quem, see 10.2.)
Even if the accusative is a person, it is connected without a preposition in Portuguese.
Eu vejo Maria.
I see Maria. (Spanish: Veo a Maria.)
Eu a vejo.
I see her.
An indirect object is connected with the preposition a in Portuguese. In English it is a little bit more tricky. If the indirect object is in front of the direct object, there is no preposition. If it is put behind, the indirect object is marked with to: John gives Maria the book. <=> John gives the book to Maria. In roman languages the indirect object always requires the preposition a.
It is useful to understand that in the case that the indirect object is a noun, it is marked by the preposition a. In the case that it is a pronoun, it has its own form.
John gives the book to Maria.
John gives her the book.
Concerning the different cases, nominativ, genitive, accusative, dative with pronouns see 6.1.1. If the cases were marked by prepositions even in the case of pronouns, the terms nominativ, genitive, accusative, dative would be superfluous. The accusative, dative, genitive would be prepositional objects. These terms only make sense, if these special cases, the accusative (direct object) and dative (indirect object) are at least in the case of pronouns treated in a different way.
(Philosophical remark: However it makes sense to understand the logic, because in other languages we have cases. Actually any type of connection can be realised as a case, in other words by changing the form of the noun / pronoun. In latin for instance "He did it with a hammer", with a hammer would be realised with an ablative, a case we don't have any more in any language. The author admits, that it is irrelevant whether we understand the difference between a "case" and a prepositional object for learning Portuguese. However roman languages and English are the exceptions. Almost all languages, russian, Arabic, german, finnish etc. have cases, while the roman languages and English have prepositional objects.)
indirect object as a noun and as a pronoun
Vou fazer uma proposta aos homens.
I will make a proposition to the man.
Vou lhes fazer uma proposta.
Vou fazer uma proposta a eles.
I will make them a proposition.
Only in the case that a connection is realised without a preposition we have a "case". Therefore there is somekind of terminological confusion in Portuguese grammar. The indirect object is a case, when it is a pronoun, but prepositional object, when it is a noun. In all the following cases, where the object is connected to the verb by a preposition, we have objective pronouns.
He thinks of us.
They talk about him.
I do that for you.
He did it with a key.
If not any case has its own pronoun, then we need a personal pronouns that fits with any prepostion. In English the same pronouns we use for the direct object, "John sees him" or the indirect object "John gives him the book", are used as well for prepositions, "John talk about him", "John does it with him", "John thinks of him" etc. etc.. In Portuguese we have complete set of pronouns that are reserved for the use with prepositions, the prepositional pronouns, see 6.1.1..
The terminology we find in grammar books written by Portuguese native speakers have a different terminology than the one we are used to, however logical from their point of view. All verbs, those who connect an indirect object as well, are called verbos transitivos indiretos in opposite to the verbos transitivos diretos, that connects without any preposititon.
verbos transitivos diretos
Comem o bolo <=> They eat the cake.
verbos transitivos indiretos
Acredita nela <=> He believes in her.
Falaram sobre n�s <=> They talked about us.
Brigou com voc�s <=> He argued with you.
� imposs�vel agradar a todos <=>It is impossible to like everybody.
Verbs that requires an indirect object, that connects or can connect with the preposition a are therefore in the same group as the verbs that connect with any other preposition. One can find that this is coherent or not, depending on the perspective.
This author would say that the indirect object is a case on its own, because as far as the pronouns are concerned, me / te / lhe / nós / lhes, it has its own form. If we say that the indirect object is just a prepostional object, the indirect object as a special case doesn't exist any more. (The English terminology follows indeed this logic. In English we have just objective pronouns and nothing else. However in English an own form for the indirect object doesn't exist indeed.) Which terminology is useful depends a little bit on the context it is used. This author would say, that at school it makes sense to speak about direct and indirect objects, because most languages make this distinction and students should be aware that we can form cases organically, by changing the form of the word itself, or synthetically, by forming the case with a combination of preposition noun / pronoun. Understanding this facilitates the learning other languages like russian, Arabic, hebrew or whatsoever.
Less problematic is the term verbos transitivos directo e indiretos. That encompasses all verbs that connects one object without preposition, a direct object, and another one with a preposition.
verbos transitivos diretos e indiretos
oferecer uma coisa a alguém to offer something to someone
O garoto ofereceu um livro ao colega <=> The boy gives a book to his classmate..
All the verbs that needs two objects, "The boy gives a book" is as meaningless as "The boy gives the girl" are verbos transitivos directo e indirectos.
The term verbos intransitivos corresponds to the English term intransitive verbs. These verbs has no object at all.
Eu vou. <=> I go.
(From a philosophical perspective there is a problem. One can wonder why in any language only the direct or indirect object survived. Actually an endless number of cases can be imagined, it seems that finnish beats any record with 13 cases, see Finnish noun cases. Any combination of preposition and noun can be theoretically a case. In this case instead of "to Jim", "for Jim", "about Jim" etc.. we would say, to illustrate the mechanism with forms created by fantasy, Jimot, Jimrof, Jimabot or something like that. We would receive very quickly languages that are so difficult to learn, that nobody could learn them as a foreign language. The fact that only the direct and indirect object survived can perhaps be explained by the fact that the combinaison of direct object and indirect object, I said it to her, is a special combinaison that describes a very special relationship, but at the end of the day we have to except that the human brain and how it wants to verbalise reality is a mystery and will forever be a mystery. There is no way to figure out how it works.)
Object can be connected to verbs with different pronouns, for instance with de (basic meaning from / of), para (basic meaning for), em (basic meaning in), sobre (basic meaning on / above). The preposition used is sometimes completely devaluated semantically, in other word the basic meaning has nothing to do with the context it is used. (Penso nela <=>~ I think in her for instance, see 6.1.3). This however happens very often with prepositions. A painting for instance can actually hang in the wall, on the wall, to the wall etc.. and in different languages different prepositions are used. In Spanish for instance a painting hangs in the wall and not on the wall.
verbs that connects with com
Simpatizo com os que condenam os pol�ticos que governam para uma minoria privilegiada.
I sympathize with polititcians who condem politicians who govern for a priviliged minority.
Antipatizo com meu professor de Hist�ria.
I don't like my history teacher.
O diretor concordou com as propostas.
The director approved the proposals.
verbs that connect with em
A modernidade verdadeira consiste em direitos iguais para todos.
The real modernity are the same rights for all.
Eu acredito em voc�!
I believe in you.
Quando estava no colegial, sobressa�a em todas as mat�rias.
When he was at school, he showed outstanding performances in all subjects.
Acredito em anjos.
I believe in angels.
verbs that connect with
Eu agradeci o convite ao diretor.
He thanked the director for the invitation.
Minha m�e deu uma boneca � minha irm�.
My mother gave my sister a doll as a present.
Dedico meu tempo aos que precisam.
I dedicate my time to those who need it.
verbs that connect with de
A empresa precisa de mim.
The company needs me.
Eu gosto de leite com chocolate.
I like hot chocolate.
Ela n�o se lembrou do meu nome.
He doesn't remember my name.
Este � o funcion�rio de quem voc� reclamou.
This is the employee they complained about.
verbs that connects with de
Estamos torcendo para voc� conseguir seu intento.
We support you in the realisation of your goals.
Ele contava muitas mentiras para a fam�lia.
He told his family a lot of lies.
verbs that connects with com
N�o concordamos com voc�.
We don't agree with you.
Ele falou com ela.
He talked with me.
Sempre simpatizei com ela, mas antipatizo com o irm�o dela.
I always found that she is a nice person, but I don't like her sister.