There are basically 4 types of recordable CD: CD-R, CD-RW, CD+R, CD+RW. (There are also CD-RAM, but they are not much used.)
As you can use CD-R disks, it's probably best to stick with CD-R or CD-RW.
CD-R disks can only be written to once. If the disk isn't full then you should be able to add more data providing that you haven't "finalized" the disk. It sounds as if your CD writing software is automatically finalizing the disk when you have written to it. You need to finalize a disk before ordinary CD players can read it.
CD-RW disks can be written to many times. You can't actually delete files on them, but you can erase the whole disk and use it again. (Again, that's not quite true; you can format CD-RW disks with a special format that make them act like hard disks, but that's not how they are normally used.)
I'd say the best disks for your purpose are CD-RW. They're not much more expensive than CD-R and you can use them over and over again. You might also want to check your CD writing software to see if there's any way to stop it automatically finalizing disks.