Praise be to Allaah.  

Firstly: 

Note that there is not one particular part of the prayer when du’aa’ should be offered, rather there are several parts of the prayer where the scholars have stated that the Messenger of Allaah (peace and blessings of Allaah be upon him) offered du’aa’, from the opening takbeer to the final tasleem. 

It is also Sunnah to make du’aa’ after the prayer and there are several du’aa’s that we will mention below in sha Allah. 

It should be noted that the best of guidance is the guidance of Muhammad (peace and blessings of Allaah be upon him) and the best of du’aa’s are those which are in accordance with the Sunnah of the Prophet (peace and blessings of Allaah be upon him). The wording of the Messenger of Allaah (peace and blessings of Allaah be upon him) is the best of wording, because he is the most knowledgeable of all people about the Arabic language, and the most eloquent in speech. Allaah enabled him to express the most profound of meanings in few words, which is called conciseness of speech. 

It was narrated that Abu Hurayrah said: I heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say: “I have been sent with concise speech.” Narrated by al-Bukhaari, 6611; Muslim, 523. 

Al-Bukhaari said: I heard that conciseness of speech means when many meanings which it would have taken books to write before are summed up in a phrase or two, and so on. 

Based on this, if you want to say du’aa’ in your prayers at the points at which it is prescribed and recommended (mustahabb) to say du’aa’, the best of du’aa’s are those which are said in the wording of the Prophet (peace and blessings of Allaah be upon him). 

Secondly: 

If you are not able to do that and you cannot memorize these du’aa’s and dhikrs, then the best of du’aa’ is that which avoids going to extremes in using fanciful language speech, and which avoids using rhyme to the point of absurdity, and which makes the du’aa’ sincere and focused on the need in question, saying whatever is easy for you  and whatever Allaah enables you to say. 

It was narrated that the Prophet (peace and blessings of Allaah be upon him) said to a man: “What do you say when you pray?” He said: “I recite the tashahhud, then I say: Allaahumma inni as’aluka al-jannah wa a’oodhu bika min al-naar (O Allaah, I ask You for Paradise and seek refuge in You from the Fire). I cannot murmur like you and like Mu’aadh.” The Prophet (peace and blessings of Allaah be upon him) said: “It is about them that we were murmuring.” Narrated by Abu Dawood, 792; classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

Thirdly: 

With regard to du’aa’ after the salaam, what the Prophet (peace and blessings of Allaah be upon him) used to do is to say, after finishing the prayer: Astghfir Allaah, astaghfir Allaah astaghfir Allaah (I seek the forgiveness of Allaah, I seek the forgiveness of Allaah, I seek the forgiveness of Allaah). Then he would say all the dhikrs that are narrated at this time. See question no. 7646. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) said: 

There is no saheeh report to say that the Prophet (peace and blessings of Allaah be upon him) used to raise his hands following an obligatory prayer, or that his companions (may Allaah have mercy on them) did that, as far as we know. What some people do, raising their hands after the prayer, is a bid’aah (innovation) for which there is no basis. 

Al-Fataawa, 1/74. 

Ibn al-Qayyim said: 

With regard to du’aa’ after the prayer facing the qiblah or facing the congregation, this is not something that the Prophet (peace and blessings of Allaah be upon him) did at all. It is not narrated with a saheeh or hasan isnaad. As for doing that only after Fajr and ‘Asr prayers, he did not do that and neither did any of his successors (khulafa’), and he did not teach his ummah to do that. Rather this is something that some people thought it was good to do to make up for not offering Sunnah prayers after those prayers. And Allaah knows best. 

The Prophet (peace and blessings of Allaah be upon him) offered most of the du’aa’s that are connected to the prayer during the prayer, and he commanded us to offer them during it. This befits the state of the one who is praying, for he is facing his Lord and conversing with Him so long as he is praying. When he says the salaam that conversation comes to a close and he is no longer standing before Him and close to Him. So why should he not ask of Him whilst he is conversing with Him and close to Him and turning to Him, and then ask of Him when he has turned away from Him? Undoubtedly it is better to do the opposite. There is another subtle point to be made, which is that when the worshipper has finished praying, and he has remembered Allaah and proclaimed His oneness, glorified Him, praised Him and magnified Him by reciting the dhikrs that are prescribed following the prayer, it is mustahabb to send blessings upon the Prophet (peace and blessings of Allaah be upon him) after that, and to make du’aa’ as he wishes, His du’aa’ should come immediately after this second act of worship, not because it is after the prayer but because it is mustahabb for anyone who remembers Allaah and praises Him and sends blessings upon the Messenger of Allaah (peace and blessings of Allaah be upon him) to make du’aa’ after that, as it says in the hadeeth of Faddaalah ibn ‘Ubayd: “When any one of you prays, let him start by praising Allaah, then let him send blessings upon the Prophet (peace and blessings of Allaah be upon him), then let him say du’aa’ however he wishes.” 

Al-Tirmidhi said: (it is) a saheeh hadeeth. It was also classed as saheeh by al-Haakim, and al-Dhahabi agreed with him. Zaad al-Ma’aad, 1/257, 258. 

Fourthly:
With regard to the points during the prayer at which du’aa’ should be offered, we will sum this up for you as follows: 

1 –After the opening takbeer of the prayer and before starting to recite al-Faatihah. This is called Du’aa’ al-Istiftaah (opening supplication): 

It was narrated that Abu Hurayrah said: When the Messenger of Allaah (peace and blessings of Allaah be upon him) started to pray, he would remain silent for a while. I said: “May my father and mother be sacrificed for you, O Messenger of Allaah. What do you say when you are silent between the takbeer and recitation?” He said: “I say: Allaahumma baa’id bayni wa bayna khataayaaya kama baa’adta bayna al-mashriqi wa’l-maghrib. Allaahumma naqqini min khataayaaya kama yunaqqa al-thawb al-abyad min al-danas. Allaahumma ighsilni min khataayaaya bi’l-thalji wa’l-maa’i wa’l-barad    (O Allaah, put a great distance between me and my sins, as great as the distance You have made between the East and the West. O Allaah, cleanse me of sin as a white garment is cleansed from filth. O Allaah, wash away my sins with snow and water and hail).” 

Narrated by al-Bukhaari, 711; Muslim, 598. 

2 – Du’aa’ al-Qunoot in Witr 

It was narrated that al-Hasan ibn ‘Ali said: The Messenger of Allaah (peace and blessings of Allaah be upon him) taught me words to say in Witr: “Allaahumma ihdini feeman hadayta wa ‘aafini feeman ‘aafayta wa tawallani feeman tawallayta wa baarik li feema a’tayta, wa qini sharra ma qadayta , fa innaka taqdi wa la yuqda ‘alayk, wa innahu laa yadhillu man waalayta wa laa ya’izzu man ‘aadayta, tabaarakta Rabbana wa ta’aalayt (O Allaah, guide me among those whom You have guided, pardon me among those whom You have pardoned, turn to me in friendship among those on whom You have turned in friendship, and bless me in what You have bestowed, and save me from the evil of what You have decreed. For verily You decree and none can influence You; and he is not humiliated whom You have befriended, nor is he honoured who is Your enemy. Blessed are You, O Lord, and Exalted).” 

Narrated by al-Tirmidhi, 464; al-Nasaa’i, 1745; Abu Dawood, 1425; Ibn Maajah, 1178. 

This hadeeth was classed as hasan by al-Tirmidhi and others, and as saheeh by al-Albaani in Irwa’ al-Ghaleel, 429. 

3 – Du’aa’ when standing up from bowing, at times of calamity.  

This is called Qunoot al-Nawaazil. This may be recited in all the obligatory prayers depending on the situation, and the worshippers behind the imam should say Ameen. See question no. 20031. 

4 – Whilst bowing.  

The Prophet (peace and blessings of Allaah be upon him) used to say: “Subhaanak Allaahumma Rabbanaa wa bi hamdika Allaahumma ighfir li (Glory and praise be to You, O Allaah our Lord. O Allaah, forgive me).” 

Narrated by al-Bukhaari, 761; Muslim, 484, from the hadeeth of ‘Aa’ishah. 

5 – During prostration.  

This is the best of du’aa’, because the Prophet (peace and blessings of Allaah be upon him) said: “The closest that any one of you may be to his Lord is when he is prostrating, so say a lot of du’aa’ at that time.” Narrated by Muslim, 482, from the hadeeth of Abu Hurayrah. 

There are many ahaadeeth concerning this, for which we do not have room to mention them here. 

6 – Between the two prostrations. 

One should say: “Allaahumma ighfir li warhamni wajburni wahdini warzuqni (O Allaah, forgive me, have mercy on me, enrich me, guide me and grant me provision).” 

Narrated by al-Tirmidhi, 284; Ibn Maajah, 898 – from the hadeeth of Ibn ‘Abbaas; classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. And there are other du’aa’s. 

7 – After the tashahhud and before the salaam: 

the Prophet (peace and blessings of Allaah be upon him) said: “When any one of you has finished the last tashahhud, let him seek refuge with Allaah from four things and say: Allaahumma inni a’oodhu bika min ‘adhaabi jahannam wa min ‘adhaab il-qabri wa min fitnat il-mahyaa wa’l-mamaat wa min sharri fitnat il-maseeh il-dajjaal (O Allaah, I seek refuge with You from the torment of Hell and from the torment of the grave and from the trials of life and death and from the evil of the trial of the Dajjaal).” Narrated by al-Bukhaari, 1311; Muslim, 588 (this version narrated by Muslim), from the hadeeth of Abu Hurayrah. Then after that he may pray for whatever he likes of the good things of this world and the Hereafter, because of the hadeeth of Ibn Mas’ood: The Prophet (peace and blessings of Allaah be upon him) taught them the tashahhud then he said at the end: “Let him ask for whatever good things he wishes.” Narrated by al-Bukhaari, 5876; Muslim, 402. 

There are many du’aa’s which may be recited during the prayer, but we cannot quote all of them in this answer. We have referred to a few of those that have been narrated. Our advice to the questioner – and to every Muslim – is to have a copy of al-Adhkaar by al-Nawawi (may Allaah have mercy on him), which is a lengthy book. If he would like to have something shorter then he should get a copy of al-Kalim al-Tayyib by Shaykh al-Islam Ibn Taymiyah, edited by al-Albaani (may Allaah have mercy on them all). 

And Allaah knows best.