A conversation: in Messages, use Delete conversation on the conversation. You see "Conversation deleted" with Undo for a few seconds; after that it's gone from the phone's message store.
Quarantined messages go to the trash first and are deleted for good after 30 days, so a mistake can still be undone for a while.
Deleting doesn't block the sender. To stop a number writing to you, block it (see "Trusted and blocked senders").
The security history keeps a short record of past events even after a message is deleted ("This message is no longer on your phone, so only this record of the event is left."), so you can still see what happened. To clear it, see "Clearing history or quarantine".