Is the question "How to be a more careful coder" or the more general "How can I be more careful in my work?". One is much broader than the other.
let's try a few broad rules
1) Put the user's shoe on. Listen and understand what is being requested of you, and why it is imoportant to the requestor.
2) First do no harm. Does everything you didn't intend to touch work just like it did before?
3) Make personal back ups of all things to be changed BEFORE working on them. Paranoia is the basis of good programming, trusting backups of others is like asking someone else to do your tiedown in mountain climbing.
4) Users shoes again - Test the software pretending you will be able to do your job ONLY through this software. test over and over. Remember those little annoyances that would make you crazy, will make the user crazy.
5) No surprises. Pretend I am doing some work for you. When would you prefer to hear the following: