I am new to Inkscape and am currently in the process of designing my first website logo. I saved the image as an .svg file. Upon opening it (in Firefox) the image came out way too small. I've looked, but have not found a solution as to how I can save it to the correct size! At the very least I'd like to have it as both the size of the Wikipedia logo and a favicon. Any help with this could be greatly appreciated.

Welcome to InkscapeForum!

To change the size of an object, you would use scaling. There are a few ways to do that. Select the object with the Selection tool, then you can grab the arrows with your mouse and drag, and the object will get bigger. In order to keep it in the same proportions (not too tall or thin or whatever) you should hold the Ctrl key while you drag the mouse. Or you could open Object menu > Transform dialog > Scale tab, and do it that way.

Since it's a logo, you probably have several objects. So just be sure that you select everything that comprises the logo.

Let us know how it goes

Edit
One more quick comment. You might pay attention to the zoom level, to make sure you're drawing it the right size to begin with

As it's a logo for a website, you might be better exporting to PNG files - one for each size you need. You keep the SVG file as your "master" for any editing that's needed, but export a collection of PNG files to actually go on the web server.

Make sure to use File=>Export Bitmap rather than "Save As", and set the dimensions in the export dialogue. See this page for more specific details of the options in that dialogue:
http://tavmjong.free.fr/INKSCAPE/MANUAL ... Export-PNG