How to render paragraphs module in twig
First add a paragraph--your_paragraph_machine_name.html.twig to the templates directory of your theme.

Accessing your paragraph fields can be done by adding twig variables in a format similar to {{ content.field_your_field }}. 

If you want only the field value, that's a little trickier. For instance if you had a List (text) field and wanted to get the value from the selected option, you could do something like:

{% for item in content.field_select_list %}
  {% if item is iterable %}
    {% for key,value in item %}
      {% if not value is iterable %}
        {% if key == '#markup' %}
          {{ value }}
        {% endif %}
      {% endif %}
    {% endfor %}
  {% endif %}
{% endfor %}


You can always run a {{ dump(content) }} to see the structure and get the values you're looking for. Hope this helps!

EDIT: You can avoid all the silly nested loops. The above example should be written like:

{% set value = content.field_select_list.0['#markup'] %}
{{ value }}


For a Text (plain) field it might look something like:

{{ content.field_text.0['#context'].value }}


To get a URL value from a Link field I've seen suggestions like:

{{ content.field_link['#items'].0.value.uri }}


But it would have an "internal:/" prefix, so I've had better luck with something like this:

{{ content.field_link.0['#title'] %}