"We painted the interior of our house and want to put back our mezuzah. Exactly how should it be put on the doorpost, and what prayers do we say?"
The Biblical commandment "And you shall write them upon the doorposts of your house and upon your gates" from Deuteronomy 6:9 is the source for the practice of affixing a mezuzah to the doorposts of your house.
Traditional Jewish practice about mezuzot is that a mezuzah should be affixed to every door in a house except for the bathroom, even if there are many rooms in a house and even if the room has more than one door. As long as a door can be used as an entrance and exit, a mezuzah should be affixed to the doorpost.
It should be affixed to the doorpost on the right hand side of a person entering the room. It should go about 2/3 of the way up the doorpost and be affixed diagonally, with the top of the mezuzah slanting in towards the house or the entrance of the room. If the doorpost is not wide enough, the mezuzah should be affixed vertically.
Before affixing the mezuzah, the following benediction is recited:
Baruch Atah Adonai, Eloheinu melech ha-olam, asher kid'shanu b'mitzvotav v'tzivanu lik-bo-a m'zu-zah.
Blessed are You, Adonai, our God, Sovereign of the universe, who hallows us with mitzvot and commands us to affix the mezuzah.
In the Reform tradition, we may interpret the requirements for mezuzah as it has individual meaning for each of us. Therefore, you may affix the mezuzah only to the exterior doors and principal rooms of the house, if that is meaningful to you.
The Eternal One spoke to Moses, saying, "Phinehas, son of Eleazar son of Aaron the priest, has turned back My wrath from the Israelites by displaying among them his passion for Me, so that I did not wipe out the Israelite people in My passion." - Numbers 25:10-11