It sounds to me like you do not have your computer backed up at all. Tha's not good.
OK, there's two ways to do this: one is to link your two computers together via cables and placing one computer into "Target mode" Since I do not know what kind of cables your Mac needs, let me just provide a link on the process. You may already have, or need to get the cabling for this.
However, the BETTER way to do this is to get yourself an external hard drive, copy the images to the hard drive and then copy them onto your new computer. About once a week or so (more often if you've just added photos, less often if you haven't) update your photos on the external drive. You can do this manually (by date is a good way to do that) or it's easier with software. Check out ChronoSync (my favorite).
Remember: there are two kinds of hard drive users: those who've had their hard drive crash and those who haven't had their hard drive crash YET.