How to restore full/incremental backup after replacing hard drive

A short while ago, I had a hardware failure force me to replace my hard drive on my laptop. The initial hard drive size was 500 GB, but when at the store I found a 1 TB drive which was affordable. After swapping the drive out, I had a bare system -- no O/S and no installed software. I tried to access my most-recent backup file to rebuild my system as quickly as possible. I booted my system off a True Image boot disc, but no matter what I did, I received nothing but errors. I was forced to rebuild my system piece by piece which took several hours. I do not know if this is a common problems faced by other people or not. I don't know what I did wrong, but my backup was unusable. Does a procedure exist which I can refer to, when next my hardware fails? I'm afraid I do not have detailed information, as far as specific errors. I'd just like to be prepared for the next time this happens.