First make sure that your system is already patched and current. Depending on how far behind you are, this might require a reboot.

yum update

Now we need to cleanup any cruft that has accumulated in the yum cache, you can do that with

yum clean all

If you need to reboot at this point, I would do it.  A nice application exists in the repository called preupgrade. It gives you a very simple GUI that allows you to upgrade with just a few clicks. Install it with

yum install preupgrade

and then, as root, run it

preupgrade

You’ll be presented with a window that has a drop down box allowing you to select which version you would like to upgrade to.  Select Fedora 10, hit apply and wait for a few hours for the process to complete.  Your box should reboot when the ugprade is done and then you will be running Fedora 10!