With little Linux knowledge this isn't a trivial task, but let's see that if we can get you through that. Keep in mind that your system may already be damaged (due to the bad sectors) and a fresh reinstall on the new harddisk would be a better option to get a known good system.
1. Start from a live-CD
2. Partition your new disk with GParted so that it has the same partitions as the old one. Don't forget to give them the same filesystem while doing this.
3. Now it is time to copy over your files. To do that we need to mount the old and the new partitions. I can only giver more precise instructions when knowing your disk layout, so please post the output of
launched as root.