If you want to access the Sector and Industry data you will need to sign up for the sector and industry data feed found here: https://intrinio.com/data/industry
Then click on tab "Documentation" ... That will take you to this page: http://docs.intrinio.com/#u-s-sector-industry-data-feed
Once you're at the documentation pages for each feed, you can find almost everything you are looking for. For instance, in API v2 there is a link on the Sector Industry page called "SIC Indices" click that link and now you have sample data on all the industry and sector codes.
Now, if you want to find out what an individual security's Sector and Industry (SIC) code is, you will have to query the security via the US Fundamentals & Stock Prices feed via datapoint.
Let's say you want to find AAPL's SIC code, you would enter this call: https://api.intrinio.com/data_point?identifier=AAPL&item=sic
That would return 3571 and 3571 is the sector industry code for "Electronic Computers"