In modern English, 'repent'/'repentance' is almost always used in a religious context, as christian bible-pounding preachers have been screaming for centuries, "Sinners repent!!!!!" However, if you look up the definition, you probably won't see any reference to this fact.
'Remorse' is a good synonym for the noun 'repentance', but doesn't have a verb form.
For the verb 'repent' you can certainly use 'regret' and you won't sound ridiculously righteous.