In addition to the options given by @Spektre, here are a couple of other things that you can try.
If you have MS-DOS 6.x/Windows 3.x or higher, you should have MSD (Microsoft Diagnostics) available, which will give you an overview of your computer's configuration. It won't help you determine the disk interface, but it will tell you the BIOS date and hard drive capacity.
If you do not have MSD.EXE available, then you can get a rough idea of the age of the system by using
debug to show the BIOS date
You should get the date of the BIOS program, which is generally around when the computer was built. Some manufacturers didn't bother to set this (or they used a bogus value), but it's generally correct.
If your computer is a consumer-class system and was made in the 1990's or later, chances are very good that the hard drive is IDE: ST-506 was essentially obsolete by the late '80s, and SCSI was more of a server option because of the cost. As well, if your hard drive is more that 80MB it's likely IDE.
However, if you're only worried about preserving data, just copy the important files to floppy disks: it's likely the only transfer mechanism that you have right now unless you invested in a null-modem cable, and you don't have to turn off the computer to use it. Most files should fit with no issues, but you may want to consider using an archiving or backup system to reduce the number of floppies that you need (most archives support spanning, which is helpful if you have files that don't fit!)