I am a little confused by your questions, so I will try to give you more information than you need, and I hope you will understand by the end 
Display the list of categories on a page or post
If you want to display the categories on a page or post. Here is the code to do that:
Ouptut the categories on the page or post
index.markdown
---
layout: home
title: Home
categories: [zed, alpha, delta, crater]
---
categories: {{page.categories}}
Output:
categories: zedalphadeltacrater
Sort and then output the categories on the page or post
index.markdown
---
layout: home
title: Home
categories: [zed, alpha, delta, crater]
---
Sorted categories:
{% assign sortedCategories = page.categories | sort %}
{%- for eachCategory in sortedCategories -%}
{{eachCategory}},
{%- endfor -%}
Output:
Sorted categories: abc,alpha,crater,def,delta,some more,zed,
Sort the post or page categories and then do something with them
Here, we can use a for loop to add a comma to the end of each category in the list of categories in a page or post:
---
layout: home
title: Home
categories: [zed, alpha, delta, crater]
---
Categories separated by commas:
{%- for eachCategory in sortedCategories -%}
{%- if forloop.last -%}
{{eachCategory}}
{%- else -%}
{{eachCategory}},
{%- endif -%}
{%- endfor -%}
Output:
Categories separated by commas: alpha, crater, delta, zed
Working with site categories
My understanding is site.categories is a hash where each key is a category name, and the value is an array of posts in that category. In other words, it is much more complicated than an array of two values.
Also, it is important to note that site.categories and site.tags were designed for posts, not pages.
For example, on my site, I have two post categories of:
Posts
agile-in-action-podcast, tech-blog
However, for your question, I added some more categories to index.markdown, which look like this:
Pages:
categories: [zed, alpha, delta, crater, abc, def, some more]
Get a list of site.categories for posts
Take this code as an example. Notice I am telling Jekyll I want a comma-separated list of all categories from site.categories :
{% assign postCategories = "" %}
{% for category in site.categories %}
{% if forloop.last %}
{% assign postCategories = postCategories | append: category[0] %}
{% else %}
{% assign postCategories = postCategories | append: category[0] | append: ", " %}
{% endif %}
{% endfor %}
{{ postCategories }}
However, the output looks like this:
agile-in-action-podcast, tech-blog
So what gives? Where are the other categories on pages? If you want those, you need to iterate through pages.
Get all categories in posts and pages
Here is code that iterates through all posts and pages to get all the categories. You can add whatever code you want, but this will create a comma-separated list of all of them.
Warning: This is an expensive piece of code because it will iterate through the entire collection of pages and posts. Granted, Jekyll does some caching, but remember that if your site gets large, this is not an ideal approach.
Tags and categories were defined for posts, not pages. However, they can be added to pages, it is simply not part of the Jekyll collection. Therefore, the following code is a hack.
index.html
{% comment %} Get all post categories {% endcomment %}
{% for post in site.posts %}
{% for category in post.categories %}
{% unless siteCategories contains category %}
{% if siteCategories != "" %}
{% assign siteCategories = siteCategories | append: ", " %}
{% endif %}
{% assign siteCategories = siteCategories | append: category %}
{% endunless %}
{% endfor %}
{% endfor %}
{% comment %} Get all page categories and append to a list of site categories {% endcomment %}
{% for page in site.pages %}
{% if page.categories %}
{% for category in page.categories %}
{% unless siteCategories contains category %}
{% if siteCategories != "" %}
{% assign siteCategories = siteCategories | append: ", " %}
{% endif %}
{% assign siteCategories = siteCategories | append: category %}
{% endunless %}
{% endfor %}
{% endif %}
{% endfor %}
Site categories: {{ siteCategories }}
Output from my site:
Site categories: agile-in-action-podcast, tech-blog, zed, alpha, delta, crater, abc, def, some more
Conclusion
If you use the official site.categories, it is designed for posts (blog posts), not pages (like index.html), but you can approximate the use so your code looks and feels the same.
If you iterate through site.categories, you can display them and work with each individually. I put a comma next to each category in my examples, but I could have easily used markdown or HTML to turn them into URLs, lists, tables, etc.
Jekyll requires an understanding of your intent, so make sure you prefix categories with page or site, like this: page.categories or site.categories
I hope this helps!