If you can run Tiny Core from the cd on your laptop, and connect to the internet:
I suggest you install gparted.tcz using the App Browser. To do this, run the App Browser, click Connect, scroll down, and click gparted.tcz. Click Go. Don't change On Boot. (This is a different method from the guides above. Easier for new users.)
Once installed, run gparted, delete existing partitions, and set up partitions the way you want them. Create an Ext3 partition, and a swap partition, for Tiny Core. A suggested size for swap is double the ram. You may also create partitions for other operating systems, if you want to install other operating systems as well.
If you can get that far let us know.
If you can get even further, following the guides above, let us know.