Well a SIM gives the phone service, a SD card holds data saved on it. So it depends on which you are actually referring to.
If this is a SIM, the data would not be on it, it would be on the old phone. If this is a SD card, then the data will only be on it if you put it on it intentionally.
If the data is only on the old phone, you will have to access that phone to get it from there and move it to the new one. The fastest way would be to connect the old phone to a computer via USB cord and move the files you want over to an empty folder on the computer. The contacts can be saved (it is done in the contacts app) to a file (it will be a .VCF file). Which can also be moved to the computer.
Once all files are in the folder on the computer, disconnect the old phone and connect the new phone to the computer. You can then move the files over to the new phone.
NOTE: If the old phone isn't working or is damaged, then it would have to be fixed to be able to get anything off of it.