@Socks
If you want to backup only database then use:
Tools > export
only if you don't want to get in game with .sql files.
But, you have access to admin area, then :
- backup mySQL database
- backup all files
OR
if you are using cPanel, then you can do FULL backup, and it will backup all files and databases. Then, on a new host (if have cPanel) just use restore button.
Also, make sure you change NS, A and other records for a domain
@DWB, use any extender plugin or if you are editing core files make sure to backup edited files cause WP update will mostly overwrite it.Also, backup original, if you screw something up, to easily restore it
OR do as Eldon Hoke told you, to edit you CSS file.
Greetings