You should not destroy your old social security card even when you get a new one as the old one can still also be used when it has the same number.
In the event you lose the new one the old social security card will still be useful.
Instead store the old original social security card in a safety deposit box or locked safe or filing cabinet.
Anything that has your social security number is something you should retain for at least a minimum number of years.
If you do wish to dispose of your old social security card you should shred it up or burn it to prevent someone from getting a hold of it.
It is bad to throw away mail with your name on it as throwing mail away with your name on it can lead to identity theft.
Any mail that you want to get rid of with your name on it should be shredded or destroyed to prevent identity theft.
Sometimes identity thieves will go through garbage and look for mail and documents to steal your identity with.
I always shred any information up and mark out any personally identifiable information on any mail.
Even if it's junk mail that has your name on it you should always shred it or rip it up or even burn it to prevent it from getting into the hands of identity thieves.
Identity thieves have also been known to piece together torn up mail and even shredded mail to try and get information.
Burning the mail or documents if possible will prevent all of that.