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