Dear Leo,
In my parish, which is quite traditionally Eastern, the priest is either seated before the icon of Christ or else is standing.
One approaches him, makes the sign of the Cross and greets him saying, "Glory to Jesus Christ, Father!"
Then one can let him know when one's last confession was, that the penance/epitimia given was fulfilled and that since that time, these are the sins I've committed since etc. This is the simplest approach.
Afterwards, the priest will give some helpful comments and will impose an epitimia or penance intended as a remedy. He will then ask you to recite the Prayer of the Publican.
You may cross yourself three times for each of the following verses:
+ God be merciful to me a sinner!
+ God cleanse me of my sins and have mercy on me!
+ For I have sinned without number, Lord forgive me!
During this time, the Priest will pray the prayer of absolution over you, placing his stole over you, and will end with: Go in peace and sin no more.
You may reverence his hand and thank him for the Confession as you depart.
That's basically the regimen in my parish.
Alex