Let me display a listing of groups and tags, which are selected by hand. So I'd pre-define what groups and tags should be proven.
So there'd two blocks within the sidebar:
Major Categories - category 1 - category 2 - category 3 ... upto 7
Quick Links - tag 1 - tag 2 - tag 3 ... upto 20
Search for a sidebar widget. I do not think this really is default functionality, since it is essential to select the tags and categires in advance within the admin.
Why don't you make use of the Groups Widget which wordpress plugin: http://wordpress.org/extend/plugins/list-tags/