As long as you do not execute files from the old system, change your passwords, and ensure that any suspect e-mails or files are removed from your online accounts, you should be perfectly safe on the new machine or simply wiping the hard drive on your old one and reinstalling. The one thing to be careful about is that if you received any infected e-mails from the hacker or any other infected files, if you run or access those particular emails or files, then re-infection could be possible, but if you first install a good anti-virus it should help with preventing accidental reinfection.
It is also worth noting that your old computer and new computer will have the same IP address (and actually, IP addresses change periodically on most residential internet connections.) Having an IP address does nothing to compromise your system. If this were the case, every web server in existence would be hacked all the time as their IPs are all known publicly. Most likely you were infected with a trojan, a software package which runs on your computer and actually grants access to the hacker by connecting to them. This could have been installed either directly, through a compromised website, through social engineering (getting you to install it without realizing it), or through an infected e-mail.
If it is in fact a targeted cyber-bullying incident, the chances are pretty good that the attack was not that sophisticated. The smart hackers are far more interested in making money either by stealing financial information or using computers as part of a larger botnet (a network of otherwise innocent computers that they have remote control of) which they can use for more nefarious purposes, such as denial of service attacks, secure end points for other hacking attempts (against companies and governments) or simply sending SPAM e-mail that is hard to block and stop.
Overall, I wouldn't worry too much about things. Hacking is not as scary as what Hollywood makes it out to be. Hackers can't just wave a magic wand and make your computer do whatever they want without doing a lot of work to compromise it first and in almost all cases getting you to do something to allow it. Follow the recommendations about only moving data you know to be legit and deleting any suspicious e-mails or postings from your online accounts (preferably from the compromised machine) as well as installing an anti-virus to scan files that you bring over from the old computer should be sufficient to keep you safe while transitioning. And should the worst happen and something sneaks by, it is a simple enough matter to restore a computer to factory defaults, reinstall the operating system and try again. If you wipe the hard drive, it is effectively impossible for the compromise to survive. (There are technically a few attacks that can load themselves in to hardware, but I'm not sure that any of them have ever been seen outside a research lab and never in wide spread use.)
