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.
The following Index anime series follow each other sequentially:
- A Certain Magical Index (2008): adapts LN volumes 1-6
- A Certain Magical Index II (2010): adapts LN volumes 7-13
- A Certain Magical Index III (2018): adapts LN volumes 14-22
Yes, the seasons are connected, and you should definitely watch them in order. Even some people who watched the previous two series felt a bit lost with the new series because it's been so long that they'd forgotten some of the details. There's a lot of continuity to keep track of here.
If you're interested in the Railgun series as well (same universe), see also: How are “A Certain Magical Index” and “A Certain Scientific Railgun” connected?