First, let me share 3 useful wordpress plugin, WP htaccess Control, WP No Category Base, WP No Tags Base. We can use WP No Category Base,WP No Tags Base at the same time. If you do not need wordress parent category name, just active WP No Category Base; same, if you do not like parent tag name or tag base, just active WP No Tags Base.
WP htaccess Control is more powerful, there are many options for you to choose. Just have a try. If you active WP htaccess Control, you do not need to install WP No Category Base, WP No Tags Base.
Second, wp_list_categories, this wordpress function is very important for you if you need specal display of wordpress category names. Detailed usage: http://codex.wordpress.org/Template_Tags/wp_list_categories
Third, $posts_array = get_posts( $args ), this is used to show wordpress post, it can meed your personal demands also, detials: http://codex.wordpress.org/Template_Tags/get_posts
Last one, the easy way to display recent posts in wordpress, only use: