You could try to remove the internal IDE hard drive from the Performa 630. Then place the drive in an appropriate external USB enclosure, or use a "caseless" USB-to-IDE/ATA adapter. Finally, connect the USB enclosure/adapter to a modern Mac. This way you may be able to see individual files on the old hard disk. Old programs can not be run on the modern computer, but you may be able to open some files through other applications. If necessary, convert (Save As) to more useful file formats on the Performa 630 first. Also, please note that there is a certain risk of altering/damaging an older file system (and its files) when trying to read the contents of an old hard disk under a newer operating system.
There are alternative ways. One could be to use a modem (internal or external) for transfers over the phone lines. The communications section of ClarisWorks would be fine for this. The receiving computer would of course need a modem and suitable software as well. Another way could be to set up a serial null-modem connection between the Performa and a more modern computer (using conversion hardware/software, as required).
You've got quote a span of time between those generations of computers! You have several choices, none of which is cost-free:
1. You can get a SCSI-based ZIP drive that plugs into the back of the Performa. You'll also need a USB-SCSI adapter on the MacBook Pro.
2. You can get an ethernet adapter for the Performa. You'll also need to update some software on the Performa (such as Open Transport, etc.- depending on what OS is on there now)
3. You can go the modem route, as explained by others.
4. You can go for professional data transfer services, such as http://retrofloppy.com
Some details on your Performa: