The Rambam answers this in Hilchot Tshuva: there are three stages
- Regret and commitment not to sin again
- Not transgressing when facing the same situation again
(some say there are four steps and separate the first between acknowledging the sin and confession)
(1:1) - Confession
If a person transgresses any of the mitzvot of the Torah, whether a
positive command or a negative command - whether willingly or
inadvertently - when he repents, and returns from his sin, he must
confess before God, blessed be, He as [Numbers 5:6-7] states: "If a
man or a woman commit any of the sins of man... they must confess the
sin that they committed." This refers to a verbal confession.
(1:1) - Regret and commitment not to sin again
Similarly, someone who injures a colleague or damages his property,
does not attain atonement, even though he pays him what he owes until
he confesses and makes a commitment never to do such a thing again
(2:1) - Not transgressing when facing the same situation again
[Who has reached] complete Teshuvah? A person who confronts the same
situation in which he sinned when he has the potential to commit [the
sin again], and, nevertheless, abstains and does not commit it because
of his Teshuvah alone and not because of fear or a lack of strength.
For example, a person engaged in illicit sexual relations with a
woman. Afterwards, they met in privacy, in the same country, while his
love for her and physical power still persisted, and nevertheless, he
abstained and did not transgress. This is a complete Baal-Teshuvah.
See the full text of the Rambam for more details.
On your first question, R Aviner says here you do not need to ask for forgiveness or make the matter public
Q: When I was young, I would take money from my father's wallet
without permission. This now upsets me greatly. What can I do? I am
embarrassed to tell him.
A: Return the money in a way he does not
On your second question, I discussed the situation with a rosh kollel who agreed one shouldn't discuss it with the spouse and for sure not to make it public and embarrass people since this is a direct Torah prohibition. He noted that if a woman had cheated on her husband, the situation would be very different (e.g., she would need to tell him, he would need to divorce her because of this).
Of course anyone confronting these situations should ask his rav for practical advice.