When you say "add new code" it's tough to tell exactly which type of code you are referring to. Are you referring to Javascript, CSS, C#, SQL, etc? There are various ways to add each of these. 

If you're looking to add code that will update the look and feel of your site then you'll need to be able to modify the "skin" of the site. DotNetNuke refers to the designs of a site as the "skin". You'll need access to the file structure of your site and inside the root folder of the site you should be able to find the skins folder at this path Portals \ _default \ Skins. This is where most skins live.

If you're looking to create a custom module then you'll most likely need to be inside of Visual Studio while you're developing.

For both scenarios you can find videos on how to do this on the Video Library Page. I would start there and watch a few videos and you'll have it figured out in no time!

There is no way Firebug can help you make permanent changes into your css. It could be used just for temporary changes, just to have a quick look what actually changing.

> Should it be through FTP (directly on CSS file) or going to ADMIN>SITE SETTINGS>STYLE SHEET EDITOR.

You may use either of them, which one suits you most.