Lets see your RewriteRule code for a 301 redirect. There's thousands of prior examples, indeed there was a post about 5 hours ago with a simple example to follow. You'll need the [R=301,L] flags to make it work.
However, the first step is that you will need to amend the links on the pages of your site to point to the new URLs. URLs are defined in links. You need to link to the URL that you want users to see and use.
Is the physical folder name inside the server changing too, or do you simply want to use new URLs out on the web?