After you have copied all the data that you need from your old mac, erase the hard drive. This is best done in target mode (
http://docs.info.apple.com/article.html?artnum=58583). On your new mac, go to Applications > Utilities > Disk Utility, and look in it the Erase options. Select the hard drive of the older Mac, and in Erase tab > Security options. Select the 7 times erase if you want to be sure your data will be gone. That is close to unrecoverable but will take way less time than the 35 times erase. After that is done, the hard drive doesn't have any of your old data in it.