Firstly: 

What is prescribed for the worshipper in a prayer in which Qur’aan is recited silently is to recite al-Faatihah and whatever he can of Qur’aan, in the first two rak’ahs, whether he is leading the prayer or praying behind an imam, because of the report narrated by Ibn Maajah (843) from Jaabir ibn ‘Abd-Allaah (may Allaah be pleased with him) who said: We used to recite the Opening of the Book and a soorah behind the imam in the first two rak’ahs of Zuhr and ‘Asr, and in the last two rak’ahs (we would recite) the Opening of the Book. This hadeeth was classed as saheeh by al-Albaani in Saheeh Ibn Maajah. 

The worshipper – whether he is leading the prayer or praying behind an imam – can recite more than one soorah after al-Faatihah in the first two rak’ahs, because of the report narrated by al-Bukhaari (775) and Muslim (822) from Ibn Mas’ood (may Allaah be pleased with him) who said: I know the pairs that the Messenger of Allaah (S) used to put together. And he mentioned twenty soorahs of al-mufassal, two by two, in every rak’ah. 

For example, the Prophet (peace and blessings of Allaah be upon him) recited al-Rahmaan and al-Najm together in one rak’ah, and al-Qamar and al-Haaqqah together, and al-Toor and al-Dhaariyaat together, and al-Waaqi’ah and al-Qalam together. 

Sifat Salaat al-Nabi (peace and blessings of Allaah be upon him) by al-Albaani, p. 104. 

The fact that it is permissible to recite two soorahs in after al-Faatihah is also indicated by the report narrated by al-Bukhaari in The Chapter on reciting two soorahs together. From Anas ibn Maalik (may Allaah be pleased with him) who said: A man from among the Ansaar used to lead them in prayer in the mosque of Quba’, and every time he started a soorah to recite for them in prayer, he would start by reciting Qul Huwa Allaahu Ahad, until he finished it, then he would recite another soorah with it. He used to do that in every rak’ah. His companions spoke to him about that and said: You start with the soorah then you do not think that it is sufficient and you recite another. Either recite it or leave it and recite something else. He said: I will not give it up. If you wish me to lead you in prayer on this basis, I will do so, and if you do not like it I will leave you. They thought that he was the best of them and they did not want anyone else to lead them in prayer. When the Prophet (peace and blessings of Allaah be upon him) came to them, they told him the story and he said: “O So and so, what is keeping you from doing what your companions told you to do? What is making you adhere to this soorah in every rak’ah? He said: I love it.” He said: “Your love for it will earn you admittance to Paradise.” 

See: Fataawa al-Lajnah al-Daa’imah, 6/403 

Secondly: 

With regard to the last two rak’ahs, the basic principle is that only al-Faatihah should be recited in them, because of the report narrated by Muslim (451) from Abu Qataadah (may Allaah be pleased with him), that the Prophet (peace and blessings of Allaah be upon him) used to recite the Opening of the Book and a soorah in the first two rak’ahs of Zuhr and ‘Asr, and he would let us hear a verse sometimes, and in the last two rak’ahs he would recite al-Faatihah (only).

This is the basic principle, but it is permissible for the worshipper to recite a soorah after al-Faatihah in the last two rak'ahs sometimes, because of the report narrated by Muslim (452) from Abu Sa’eed al-Khudri (may Allaah be pleased with him), that the Prophet (peace and blessings of Allaah be upon him) used to recite in the first two rak’ahs of Zuhr prayer approximately thirty verses in each rak’ah, and in the last two rak’ahs approximately fifteen verses, or he said: half of that. In the first two rak’ahs of ‘Asr he used to recite in each rak’ah approximately fifteen verses, and in the last two rak’ahs approximately half of that. 

Shaykh al-Albaani (may Allaah have mercy on him) said: This hadeeth indicates that reciting more than al-Faatihah in the last two rak’ahs is Sunnah, and many of the Sahaabah held that view, including Abu Bakr al-Siddeeq (may Allaah be pleased with him). This is the view of Imam al-Shaafa’i, whether that is in Zuhr or any other prayer. Among our later scholars, this view was followed by Abu’l-Hasanaat al-Laknawi… end quote. Sifat al-Salaah, p. 113. 

Shaykh Ibn ‘Uthaymeen was asked: 

If the worshipper has finished reciting al-Faatihah and another soorah in a prayer in which Qur’aan is recited silently, and the imam has not bowed, should he remain silent? 

He replied: 

The worshipper should not remain silent if he finishes reciting al-Faatihah and another soorah before the imam bows, rather he should recite until the imam bows, so if he is in the two rak’ahs after the first tashahhud and he finishes reciting al-Faatihah but the imam does not bow, he should recite another soorah until the imam bows, because there should not be any silence in the prayer except when the worshipper is listening to the recitation of his imam. End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen, 15/108 

And Allaah knows best.