Most often Mass continue with the Penitential Act after the greeting. Note there is a formula for confession, and also an absolution. The 2000 edition was amended with the last sentence of 51a to remind readers that this absolution isn’t quite what the ordinary form provides:
51. After this, the Priest calls upon the whole community to take part in the Penitential Act, which, after a brief pause for silence, it does by means of a formula of general confession. The rite concludes with the Priest’s absolution, which, however, lacks the efficacy of the Sacrament of Penance.
Sometimes the Penitential Act will be replaced:
From time to time on Sundays, especially in Easter Time, instead of the customary Penitential Act, the blessing and sprinkling of water may take place as a reminder of Baptism.[Cf. The Roman Missal, Appendix II]
Some interesting material on the Kyrie Eleison:
52. After the Penitential Act, the Kyrie, eleison (Lord, have mercy), is always begun, unless it has already been part of the Penitential Act. Since it is a chant by which the faithful acclaim the Lord and implore his mercy, it is usually executed by everyone, that is to say, with the people and the choir or cantor taking part in it.
Each acclamation is usually pronounced twice, though it is not to be excluded that it be repeated several times, by reason of the character of the various languages, as well as of the artistry of the music or of other circumstances. When the Kyrie is sung as a part of the Penitential Act, a “trope” precedes each acclamation.
This piece may or may not be considered part of the Penitential Act. If not, it would be one of those musical elements which stands as a rite on its own.
This may be a choral piece, but “usually” is a matter for assembly participation. My interpretation would be that the usual experience of any worshiper would be to participate, not just attending to a majority of Masses offered.
There is considerable leeway for the multiple repetition of invocations, rather than just the common call and response. Composers or arrangers are given room for “artistic” expression.
Thoughts?
Catholic Sensibility 
Can a person who has not been to confession go to communion after he has said the penitential prayer if he plans to go to confession in the near future? I have been told that this prayer can be said and that you can go to confession at mass.
Liz, Catholics are obliged to confess serious sins before receiving Communion. Participation in the penitential rite, ideally, is sincere and reflects an actual personal contrition. It is a common understanding that venial sins are forgiven in this way, but it is not the primary purpose of the rite to forgive sin. The sacrament of reconciliation is still the optimal route.
Optimal infers not only route