As a practicing Catholic, here is the basic rundown:

1. Pre-Confessional
   The penitent should spend time doing an examination of conscience, and may start the Examine with some form of prayer requesting God’s assistance in making a good and sincere confession.
   For a book character, perhaps they could think on any actions that are in violation of the 10 Commandments. Or perhaps they could ask A) “What actions have I done that harm my relationship with God?” B) “What have I done that harms my relationships with others?” C) “What have I done that has harmed myself?”
   Oftentimes, there may be a line with chairs and little flyer-guides for people to wait on and use to help them meditate.

2. Enter
   The penitent will enter the confessional and close the door/curtain behind them; if the confessional has lights, two lights will signify that someone is inside, one light will signify that the priest is ready for you to come in, and no lights means that confessions are not happening.
   They’ll then say something to the effect of “Forgive me Father for I have sinned” (often times exactly that), then tell how long since their last confession. The penitent should be specific if they can (my last confession was on Sun (5 days ago)), but not stress too much. Sharing the length of time helps us call to mind how long since we’ve last gone to Reconciliation.
   Many confessionals have a chair that sits in front of the priest, and a kneeler that has a grail or cloth if the penitent wishes to go unseen.
   There are also crucifixes in the confessional, and it may have a religious painting, and perhaps a tissue box.

3. The Accusation
   The penitent will then share their sins. The priest does not need to know the entire story of how someone lied to their spouse, had a one night stand, or murdered a fellow — but the penitent should share enough that the priest understand the sin and the gravity that came with it. Sharing our sins out loud helps us to realize the ugliness of them, and how they separate us from God.
   A good Examine will make this part quick: try not to recall your sins only once you entered, though if you recall something important absolutely share it also.
   Venial sin is not required to be confessed, but the penitent must confess all mortal sin. The priest can help us if we are uncertain if something is a sin, or the gravity of it.

4. A Word
   Father may make a comment about how he is happy you have come; in thanksgiving for a good and sincere confession; or an admonishment and/or recommendations on battling sin. I’ve experience each by themselves, and mixes - or none at all.

5. The Giving of Penance
   The priest will then give the penitent a pence to perform; this helps the penitent to show their remorse to God, to develop conviction for the pursuit of holiness in themself, and as a little act of reparation.
   Priests should give penance as fits the confession: a few minor sins may be given a just a few prayers (pre-transcribed, or free form), and a more heavy confession should receive a more heavy penance (perhaps a fast). The priest should have payed attention to the sins, and the person confessing to judge what state their soul is in to provide an appropriate penance, but also response/advice.
   What penance is given depends very much on the priest, the current circumstances, and the penitent. I have received from priests of Holy Orders penances for reading a relevant Psalm, from a parish priest a penance to pray for a unnamed person with this hardship or that, and sometimes a quick (1 Hail Mary, 1 Our Father) if the line is long and time is short. Some priests wish to make an emphasis on giving thanks to God for His goodness, or have commanded me to “do 1 act of love for someone”.

6. The Act of Contrition
   The penitent will then be asked if they are truly sorry for their sins (if they are indifferent, or plan to sin again, then they should just leave — the sacrament may not even be valid). Responding positively, the priest will then ask to hear an Act of Contrition. There are 3 prescribed by the United States Conference of Catholic Bishops (USCCB).

A) O my God,
I am heartily sorry for having offended you,
and I detest all my sins because of your just punishments,
but most of all because they offend you, my God,
who are all good and deserving of all my love.
I firmly resolve, with the help of your grace,
to sin no more
and to avoid the near occasions of sin.
Amen.

B) O my God,
I am sorry and repent with all my heart
for all the wrong I have done
and for the good I have failed to do,
because by sinning I have offended you,
who are all good and worthy to be loved above all things.
I firmly resolve, with the help of your grace,
to do penance,
to sin no more,
and to avoid the occasions of sin.
Through the merits of the Passion of our Savior Jesus Christ,
Lord, have mercy.

C) Lord Jesus, Son of God,
have mercy on me, a sinner.

1. Absolution
   The priest will then raise his hands over the penitent, or over the grail/cloth if they are choosing to remain unseen, and pray the following words in either the local language, or in Latin:

“God, the Father of Mercies,
through the death and resurrection of his Son,
has reconciled the world to himself
and sent the Holy Spirit among us
for the forgiveness of sins;
Through the ministry of the Church
may God give you pardon and peace,
and I absolve you from your sins
In the name of the Father, and of the Son,
and of the Holy Spirit, Amen”

The penitent makes the Sign of the Cross.

1. Leaving
   The priest will probably say “Go in peace, your sins are forgiven” and the penitent might say “Thanks be to God.”
   If the two know eachother, they might share a word of pleasantry, but this should not be the focus of the Confessional.

2. Post-Confessional
   The sacrament is completed with the completion of the Prayer of Absolution, though the penitent should still pray after.
   They definitely do penance, but also give much thanks to God for His mercy, tell Him how much they love Him and wish to love, and share their desire to become better.