I strongly recommend that you do not try to transfer your old programs and such to your new computer. Instead, download new copies of the software from trusted sources, find new replacements for really old software, or contact the vendor and ask them if they can provide you with a new installation for previously purchased software. Otherwise, you will probably just reinfect your new computer and all of the old problems will come creeping back into your life.
Do not allow the infected computer to remain on your network. It may seem harmless to let it stay connected, but all this does is turn your private network into a hostile public environment. This infected machine can easily sniff all incoming/outgoing traffic across your home network and continue to steal passwords, personal information, etc. Even if you don't allow this machine to share files with other computers, it can still spy on your communications. Heck, in this scenario, this machine could easily spy on your HTTPS communications using something like SSLStrip.
Once you take this infected machine offline, you should copy your important data to a CD, DVD, thumb drive, external hard drive or something similar. You also want to make sure to disable auto play on your new computer so you don't accidently reinfect yourself. The infected machine will most likely infect the media and hope the new computer will auto play hidden nasties. Once you have successfully transferred over the data, if you used a thumb drive or external hard drive, I would just reformat the device to be extra careful. Once formatted, you can create a new partition and use it to store your clean backups.
Copying over Microsoft Office (or similar) documents can be just as dangerous as executables. Be extremely careful when backing these kinds of files up. I understand the need to transfer over these types of files, but use caution when doing so. If you do copy these types of files over, make sure to scan them on your new computer before opening anything. While this won't detect every threat out there, it would at least stop you from reinfecting yourself with known malware.
Stay away from Microsoft's Easy Transfer junk. If you go this route, you will most likely just reinfect your new computer. This scenario isn't your typical "migrate to a new computer", because the old computer has been compromised and you can't trust it. Software like Microsoft's Easy Tranfer is great for typical migration, but does a terrible job in situations like this, because you really need to only move over important documents and manually recover old software, etc.
Edited by Kaosu, 28 June 2014 - 03:35 PM.