Wuḍoo' (Ablution)
Adhān (The Call To Prayer)
The Second Rak'ah (Unit Of Prayer)
The Middle Tashahhud (Testification Of Faith)
The Third And Fourth Rak'ah (Units Of Prayer)
The Final Tashahhud (Testification Of Faith)
Remembrance Of Allah After Praying
Witr Prayer
Jumu'ah (Friday Congregation)
Taraweeḥ Prayer (Optional Night Prayers During Ramaḍaan)
'Eid Prayer
Prayer For Seeking Guidance
Janāzah (Funeral Prayer)
Reciting Al-Fātiḥah

After saying bismillāh, you should begin reciting Sūrah al-Fātiḥah (the opening chapter of the Qur'an). 

There are some prayers in which the person praying on their own, or the person leading others in prayer is recommended to recite the Qur'an alound, and there are other prayers in which it should be recited quietly. As for the person who is following another in prayer, their recitation is always done quietly.

The prayers in which the Qur'an is recited aloud are the first two rak'ah (units) of Fajr(the dawn prayer), Maghrib (the sunset prayer), and'Ishaa' (the evening prayer), as well as Jumu'ah (the Friday congregational prayer), the 'Eid prayer, the prayer for rain, and the prayer at the time of an eclipse of the sun or moon. It is also common to recite voluntary night prayers aloud.

The prayers in which the Qur'an is recited quietly are Dhuhr (the noon prayer), and 'Aṣr (the mid-afternoon prayer), and the voluntary prayers, other than the night prayer. It is also recited quitely in the third and fourth rak'ah of Maghrib (the sunset prayer), and 'Ishaa' (the evening prayer).

The Prophet (may the peace and blessings of Allah be upon him) said: 

"Truly, the one who has one of the finest voices among the people for reciting the Qur'an is the one whom you think fears Allah when you hear him recite."  [Ibn Mājah: 1339

The Prophet (May the Peace and Blessings of Allah be upon him) also said:

"Beautify the Qur'an with your voices." [Abū Dāwood: 1468 , an-Nasā'ī: 1015, Ibn Mājah: 1342]

You should recite Sūrah al-Fātiḥah in full, as this is an essential part of the prayer, without which the prayer is not valid. It consists of 7 verses:

بِسْمِ اللَّهِ الرَّحْمَنِ الرَّحِيمِ
ٱلْحَمْدُ لِلهِ رَبِّ الْعَـٰلَمِينَ
ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ
مَـٰلِكِ يَوْمِ ٱلدِّينِ
إِيَّاكَ نَعْبُدُ وَإِيَّاكَ نَسْتَعِينُ
ٱهْدِنَا ٱلصِّرَٰطَ ٱلْمُسْتَقِيمَ
صِرَٰطَ ٱلَّذِينَ أَنْعَمْتَ عَلَيْهِمْ غَيْرِ ٱلْمَغْضُوبِ عَلَيْهِمْ وَلَا ٱلضَّآلِّينَ


bismillaah hir-raḥmaan nir-raḥeem
alḥamdu lillaahi rabbil-‛aalameen
maaliki yawm-id-deen
iyyaaka na‛budu wa iyyaaka nasta‛een
ṣiraٰṭ-alladheena an‛amta ‛alayhim ghayril-maghḍoobi ‛alayhim walaḍ-ḍaaalleen


In the name of Allah, the Most Merciful, the Bestower of Mercy
All praise is for Allah, Lord of the worlds.
The Most Merciful, the Bestower of Mercy
Owner of the Day of Recompense
You (alone) we worship, and You (alone) we ask for help
Guide us to the Straight Path
The way of those upon whom You have bestowed Your Grace, not of those who have evoked [Your] anger, or those who went astray.

On finishing the recitation of this chapter, you should say Ameen (an Arabic term meaning 'Accept our supplication!'), as the Messenger of Allah (may the peace and blessings of Allah be upon him) used to say this at the end of reciting the al-Fātiḥah [al-Bukhārī: 782, Abū Dāwood: 932].

The Prophet (may the peace and blessings of Allah be upon him) said, 

"Whoever does not recite Al-Faatiḥah in his prayer, his prayer is invalid."  [al-Bukhārī: 756

Those who have not memorised al-Fātiḥah should say in its place:

"Subḥaan Allaah, wal-ḥamdu-Lillaah, wa laa ilaaha il-lallaah, wa Allaahu akbar, wa laa ḥawla wa laa quwwata illa Bil-laah."

This is because the Prophet (may the peace and blessings of Allah be upon him) said to the one who told him that he could not read al-Fātiḥah: 

"If you know some of the Qur'an, then recite it; otherwise say: Subḥaan Allaah, wal-ḥamdu-Lil-laah, wa laa ilaaha il-lallaah, wallaahu akbar, wa laa ḥawla wa laa quwwata illa Bil-laah" [Abū Dāwood: 832

Which translates as:  

How far from imperfections is Allah, praise be to Allah, there is no deity worthy of worship but Allah, Allah is the Most Great, and there is no strength nor power except with Allah.

This of course, does not mean, that if the person is unable to read it, then he or she would recite these words for the rest of their lives, but they should make utmost effort to learn it by heart, and these words are a temporary replacement.

The guidance of the Prophet (may the peace and blessings of Allah be upon him) is to recite it one verse at a time, and to pause at the beginning of each verse. This is how the Prophet (may the peace and blessings of Allah be upon him) used to recite the whole chapter, pausing at the end of each line and not joining one line to the next, even if there is continuity of the meaning.  

[at-Tirmidhī: 2927]