This is the site I want to gather data from but I'm really not sure where to start.
You can use python, R or other code language as you prefer:
Example for python: https://stackoverflow.com/questions/6325216/parse-html-table-to-python-list
Here just two short examples from stackoverflow. If you want to iterate the process for various tables you can use a for cycle.
In python there are many libraries available for web scraping these are some of the important ones scrapy, Beautiful Soup etc. In php we have simple_html_dom.php which can parse data from websites. You need a good programming skills to scrape websites by these programmes.
The best way for non-programmers would be using some online web scraper such as import.io but let me warn you not many sites work perfectly with this tool.