I don't know if I can be of any assistance but here goes....
First off, consider using cfdisk. Not that there's anything wrong with using fdisk but if you're a newbie to all of this, cfdisk is easier to use (looks and feels a lot like fdisk from DOS) and since you're a newbie, it will be easier to set things up correctly...
So pass the command 'cfdisk' when you've booted with the CD and logged in as root.
Set your partitions as you please. Personally i use 1 partition for the root ( / ) about 12 GB and one for /home, about 120 GB (the large /home is due to an FTP-server). The space is optional but say you have a 20 GB drive, you should set / to about 12 GB and /home to 8 GB. You get the point.
Ah! Almost forgot the SWAP-space. Personally i removed my swap when I upgraded my computer, I was told that if you have 512 MB DDR-RAM or more the swap is obsolete. But once again this is optional, if you need/want a swap-partition, set it to about double the size of your RAM...
As for the error during install I would personally recommend taking the disc out and wiping it with a dry cotton cloth from the center moving out in a straight line. If that doesn't work rince the disc with water (should be finger-warm). Administer soap in a thick layer to all of the disc and rince it once again with water until all the soap is gone and wipe it off with a cotton cloth in the same way as before.
If the disc still refuses to comply, do what hackers call a "brute-force-attack" meaning:
1. Take the disc out of the CD-ROM
2. Pound it to the floor
3. Take a sharp object and run it over the CD a repeatedly
4. Put it in the microwave (this is extremely cool btw
5. Burn it with a blowtorch
6. Bury it in your backyard/backyard of choice
7. Buy a new one
There you have it, good luck, and be sure to post your progress....