How do I display entries 1 by 1? I need to display entries that contain an image, and a piece of text, on the home page of my website. 

When someone clicks 'next', I would like it to go to the next entry (by post date) and display that entries image and text. I also need the user to be able to click on 'previous' (by post date again) and 'random'. 

How do I go about this? 

Thanks!
You could use the getNext() and getPrev() methods on your entry like this:

{% set params = craft.entries.section('yoursection').order('postDate asc') %}

{% set prevEntry = entry.getPrev(params) %}
{% set nextEntry = entry.getNext(params) %}
{# to access url, I use #}
{{ prevEntry.url }}
{# to access title, I use #}
{{ prevEntry.title }}


NOTE: I've shamelessly pinched this code from the answer here.

You need to setup the params variable so that it would create a list of entries which would include your current entry and the next/prev entries you expect. So include your ordering by postDate etc.