To upgrade without breaking things
You must be administrator on the computer that you're doing this on
I recommend going into settings and changing brightness & lock>turn screen off to never, and go into settings>power>suspend when inactive for... to Do Not Suspend, just so your system doesn't break itself automatically.
I personally find using the command line easier, because I know what my system is doing. To go that path, open up a terminal and enter
sudo apt-get update; sudo apt-get upgrade; sudo do-release-upgrade, type in your password (it will not show up, THIS IS NORMAL), followed by enter. Whenever it says something along the lines of "Do you really want to upgrade [y/N]" type y, followed by enter. When it says "config file path/to/file is different from the package maintainer's version", press enter. Once your system goes through the upgrade process, you should be prompted to reboot, type y, then your system reboots into 13.04.
If you want to use the GUI:
Start up the update manager, then see if it wants to upgrade. If it doesn't mention upgrading to a new version of Ubuntu, you can force it to check for new ubuntu versions by going into a terminal and typing
update-manager -c, then selecting to upgrade to a new version. This process will be simpler than using the terminal, as there are much better onscreen instructions, but it usually doesn't tell you anything useful if it breaks.
Ubuntu 13.04 has much better performance, and it's just at the start of its support cycle, so you won't need to upgrade again for a while, and due to release management, the software on it will be newer versions than are ever available on 12.10.