Cable: yes, if both PC's have an Ethernet-connection,
Wireless network (using a router or switch): yes, if both PC's can be connected to it.
Using a router makes sense if you want to connect both PC's to the Internet simultaneously.
Alternatives might be:
- use a USB-stick (depends on the amount of data), it isn't really practical to transfer 100 Gb of data using a 1 Gb stick, even if all files are smaller than 1 Gb.
- use an external hard disk (250 Gb for $125)
- burn to DVD-R or DVD-RW
All of these are viable transport media.
I understand from your question that you didn't yet regularly BACKUP your data. If you did such, you could just restore everything you want from the backup of your old computer to your new computer. That's taking a BIG risk of losing your data, not having a backup on a reliable other medium than just one hard disk.
I really advise you to start making backups, unless - of course - you really don't mind losing all your data. But few people do think so.
Hope this helps.
Kees