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 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?
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.