When I comment on a post it does not display on my timeline or News Feed
Regardless of this, I thought all posts to public pages would show up on one of my pages.
There are two aspects here: posts and comments.
Posts to public pages, for example, if you post on the public page of your local cinema, will be publicly viewable by anyone with access to that page (which is everyone) regardless of your privacy settings. Posts you're tagged in and things that you post on your timeline are currently only visible to you, but otherwise would be shown to all of your friends in their News Feeds and on your own timeline.
Think of it this way: when you post on your own timeline (which can be done by visiting your profile page, or through your News Feed) it is like writing in your diary. Your privacy settings decide who can read your diary, so that might be just you, or all of your friends, or all of the public. You can control the privacy settings of your own diary.
If you post on your friend's timeline, you are writing in their diary. Everyone who they have allowed access to their diary can see what you posted.
If you post on a public page, it is like writing on their billboard. The public can all see it.
Moving on to comments: If you comment on a friend's page or a public page's post, that comment is visible to anyone who could view the original post, but it won't show up on your timeline, because the original post wasn't written in your 'diary'.
If you want to see the history of all your actions, you can visit your activity log via your profile page, it should be at the bottom right of your cover photo.