How to set and save the sent information in Samsung mobile phone s4?

According to the situation you described, the default SMS of Samsung S4 is chat box mode, which will be automatically saved after you send the message.

If you need to back up SMS, you can try:

S4 SMS is backed up to the SIM card of the mobile phone. Please follow the following steps: enter "information" and press a short message to make a backup. A dialogue window will pop up and be copied to the SIM card! Note: If the backed-up short message exceeds 70 bytes, it cannot be backed up to the SIM card, nor can the multimedia message be backed up.