Real Time Clock DS1307 interfacing with Arduino Uno
From the code:

second = (byte) ((Serial.read() - 48) * 10 + (Serial.read() - 48));
minute = (byte) ((Serial.read() - 48) *10 +  (Serial.read() - 48));
hour  = (byte) ((Serial.read() - 48) *10 +  (Serial.read() - 48));
dayOfWeek = (byte) (Serial.read() - 48);
dayOfMonth = (byte) ((Serial.read() - 48) *10 +  (Serial.read() - 48));
month = (byte) ((Serial.read() - 48) *10 +  (Serial.read() - 48));
year= (byte) ((Serial.read() - 48) *10 +  (Serial.read() - 48));


All have two characters except for dayOfWeek

So for right now (2014-01-25 13:58:00) it would be 14 characters (including 'T')

T0058136250114


You only have 13 characters in your command.

Also you need 4K7 - 10K pullups on SDA and SCL. Note that "Command: 13" in the log is just the carriage return character (enter). Since you had too few characters in the command, it was interpreted as part of the command.