What you need is a hard-drive caddy, something like this: Caddy. But you will have to make sure that you get the right type to suit your hard-drive connection, either the old IDE or the new Sata. I suspect that you will have the IDE. You can actually get a IDE/Sata caddy which might be your best bet, THIS. Pop your hard-drive in the caddy use the usb cable to connect to your new PC then all you have to do is copy the photos etc to the new PC and of course you will have a back up on the old hard-drive. Though it would probably best to,once you have copied the files across you then wipe the old hard-drive clean and then copy all your stuff back to it, and then you will have a nice clean backup.
Another way would be to add your old drive as a slave to the new one then copy across but again if your old drive is IDE and your new motherboard does not have an IDE connection on it,and most of them don't these days,then this will be a problem.
I know it sounds a lot of faffing about but it is not really and you will get plenty of help here.