Home is laid out in three columns. The middle one is the feed.
The feed is built from two sources and nothing else: the people you have as peers, and the communities you belong to. You can post to your own feed at any time. Posting to a community’s feed is the part that needs membership.

Title, opening lines, sometimes an image, then the author. The line worth noticing is the one beneath the author’s name.
“For [network]” means the post was published to that community. No such line means the author published it to their own feed.

They are the same four on every screen.

The share icon opens four options: Copy link, Share on your feed, Share on Facebook and Share on LinkedIn.
Share on your feed is the one to know. It puts someone else’s post in front of your own peers, with the original attribution intact.

It is built from your peers and your communities, so it fills as those do. The suggestions in the right column are the quickest way to start.
It is Like. The number beside it is how many people have marked the post as worth reading.
Saved items, in the left column here and on your profile. Everything you bookmark lands there.
Open that community and use its own feed. This page deliberately mixes everything you are connected to.
On your profile under My feed, and in the feed of any community that accepted them. Manage my posts shows all of them with their status.