Facebook news feed is not showing most of the current liked pages updates and posts

Facebook keeps on performing some sort of A/B testing, where few users (based on some business logic of facebook) will see one UI, while rest will see other UI.
I don't know how it could be possible at the time this question was posted, but currently you can give a try to these steps:

1. On sidebar of the homepage, you will see 'triple dot' next to News Feed link.
2. Click on this. If you don't have this option, it means right now this feature is not available to you.
3. You will see multiple options here, like:
   i. Prioritize who to see first: Select names of friends whose posts you want to see on the top of your timeline.
   ii. Unfollow people to hide their post: You can unfollow your friends right from here, so that their post will never be displayed on your TL, unless you follow them again.
   iii. Reconnect with people you have unfollowed: If you had previously unfollowed someone, you can refollow them right from here.
   iv. Discover Pages that match your interest: Relevant pages based on your activity will be suggested by facebook. Click like to get their notification.
   v. See more options: If you have any hidden apps then you can manage them from here.
   
   Please have a look at the screenshot:
   

You can manage your pages by going to your 'Profile' and clicking on 'Info' on the left.
It lists out all the pages you have 'liked' sorted by category. There is a edit link on the right over there. You can remove the pages which you don't like over there.

Regarding how to show updates from the pages you've liked, I think facebook automatically hides updates from the pages you don't interact with.