You should create labels for the categories you want, then click the archive button to move it to the All Mail folder (out of the inbox). E-mail that is archived will not show up in the inbox.
Think of it this way... Gmail works on labels, not folders.
There are a few default labels already in place...
1) The Inbox label - shows you mail that is unread and Not archived.
2) The Sent Mail label - shows you e-mail that you have created and is automatically moved to archive as well.
3) The Drafts label - shows you mail that is only under drafts and not in any other label yet (also appears in All Mail).
4) The All Mail label - shows you mail that is in any label. This is the universal label. It shows everything (Except Spam and Bin).
5) The Spam label - shows you mail that has been flagged as spam, does not appear under any other label. Gets deleted after a month.
6) The trash Bin - shows you mail that has been marked for deletion. Gets permanantly removed after a month.
It is not necessary to make a label for "saved e-mails" ... everything in the All Mail folder is already saved - so you are duplicating the label there. Just select the e-mail and click the archive button to move it to the All Mail label.