WordPress rewritten URL structure

My WordPress site can be broken in two parts:

1. Content type as pages, having URLs like: site.com/page/subpage/
2. Content type as posts, having URLs like: site.com/category/postname/.

The URL site.com/blog/ is a page having a “Blog posts” template. If I open one of the listed posts, I get redirected to site.com/category/postname, but I’m interested to have /blog/ in front of it, like the following: site.com/blog/category/postname. Also, if I try to access a blog post without /blog/ in front, I want to be redirected to the correct URL.

Another thing, the archive is of the form site.com/01/2013/, but I would like to be like: site.com/blog/archive/01/2013.

My question is: how can I rewrite the URLs so that I can those bolded above; to have /blog/ in front of the /category/ and /blog/archive/ in front of /01/2013/