Are the "A Certain Magical Index" seasons connected? Can you watch A Certain Magical Index season 3 without watching the other seasons first? Do you have to watch them in order? Are all the Index seasons connected or is season 3 a new story?
The seasons of the anime adapt the light novel series which tells a single, continuous story following the same set of characters across many volumes. As such it's not recommended that you start with season 3 of the anime, since it assumes you already know who the main characters are and what major events already transpired in the previous story arcs. 

Starting with season 1 is generally recommended for any sci-fi/fantasy series with a long, complicated narrative like Index. Missing even one episode can often mean missing crucial information that helps piece together character motivations and plot developments.