After the takbeer is performed, and you have begun the prayer, there are a number of different supplications that you can recite. The recitation should be done quietly, so that you are the only person who can hear what is being said. The most famous of these supplications narrated from the Prophet (may the peace and blessings of Allah be upon him) is the following:

“سُبْحَانَكَ اللَّهُمَّ وَبِحَمْدِكَ وَتَبَارَكَ اسْمُكَ وَتَعَالَى جَدُّكَ وَلَا إِلَهَ غَيْرُكَ”

Transliteration: "subḥaanak-allaahumma wa biḥamdik, wa tabaa-rakas-muk, wa ta‛aalaa jadduk, wa laa ilaaha ghayruk"

Translation: How far from imperfections You are, O Allaah, and I praise You. Blessed be Your name and exalted be Your Majesty, and there is none worthy of worship except You.

[Abū Dawood: 775, an-Nasā'ī: 900, Ibn Mājah: 804, at-Tirmidhī: 242]

The Prophet (may the peace and blessings of Allah be upon him) also used to read the following supplication:

“اللَّهُمَّ بَاعِدْ بَيْنِي وَبَيْنَ خَطَايَايَ كَمَا بَاعَدْتَ بَيْنَ الْمَشْرِقِ وَالْمَغْرِبِ ، اللَّهُمَّ نَقِّنِي مِنْ خَطَايَايَ كَمَا يُنَقَّى الثَّوْبُ الأَ بْيَضُ مِنَ الدَّنَسِ ، اللّهُمَّ اغْسِلْنِي مِنْ خَطَايَايَ بِالثَّلْجِ وَالْمَاءِ وَالْبَرْدِ”

Transliteration: "allaahumma baa‛id baynee wa bayna khaṭaayaa-ya kamaa baa‛adta baynal-mashriqi wal-maghrib, allaahumma naqqinee min khaṭaayaa-ya kamaa yunaqqath-thawbul-abyaḍu minad-danas, allaahummagh-silnee min khaṭaayaa-ya bith-thalji wal-maa’i wal-barad"

Translation: O Allah, distance me from my sins just as You have distanced The East from The West, O Allah, purify me of my sins as a white robe is purified of filth, O Allah, cleanse me of my sins with snow, water, and hail.

[al-Bukhārī: 744, Muslim: 598]

Either supplication can be read as both are proven from the sunnah. There are also other supplications which you can find on our partner site, duas.com.