Get relationships between tables when no foreign keys are present
You have 3 possible options, but you most likely want to start with #2 and combine it with #3.


Use php to print out the Entity Metadata Wrapper for various nodes objects of different types. 
Use the Devel Module and view the Node object
Use Views


If the Entity definitions were properly implemented (a huge IF) it is possible to print out the relationships using entity_metadata_wrapper from php.  You can even write it out to a file from command line using Drush but this will take some advanced skills.  However, this typically results in tens of thousands of lines, an overwhelming amount of information, with many circular references.

Using the Devel Module will be much easier as you can simply click down into what the Node object is presenting.  This may not show all the fields if they aren't filled out so you will want to create a test Node of each type with data in each field. If filled properly this may help relate structures when looking at the raw MySQL data as well.

Using Views, Drupal's built in query builder, will help you either understand the schema or just export it by adding the Views Data Export module. You will want to start from a perspective of particular Node type (a content type).  Each Node type will show fields it can access and you should be able to build the queries from there.  You may need to add Relationships (joins) to get all the data depending on how the developers built the system, its hard to say. Drupal will limit you to only the tables that are defined with Relationships. 

You can see the MySQL Queries you are building at the bottom of the page if Show the SQL Queries is enabled in the Views settings tab (/admin/structure/views/settings).