I am new to Inkscape and have a limited knowledge about all the different files. My question is how should I save my logo to email, post on my site, use for printing shirts and banners etc.

Welcome to InkscapeForum!

For email, it depends on what kind of software the recipients have. Not all image viewers can display SVG files. So if you happen to know that the recipients will be able to view the SVG format, then you could use that. Otherwise, something more common would probably be better, and PNG is darn near universal (if not entirely!). To save a PNG version of your Inkscape image, you should use File menu > Export Bitmap (and not Save as Cairo PNG).

For posting on a website, the considerations are similar to email. Not all browsers can display SVG images (any IE version before 9 cannot display them at all, but IE9 and most other browsers can). (It's my understanding that no browsers can display all SVG images, so depending on what's in the image, even with a supported browser, some SVG images still won't be displayed.) So if you need to be sure that everyone who visits the site/page can see it, you should use PNG.

For printing shirts and banners, you'll need to ask the printing company that you hire, what kind of file format they need. I guess there are different ways of printing, and people tend to work with whatever they feel comfortable using. The ones I've heard the most are EPS and SVG. But you should definitely ask them what they need