I know about Chess960, and you can play that online at lichess.org. Furthermore, I like playing Go as well. But are there any good variants and other ways to train that you use, and you know to work well?
Wikipedia has a pretty complete listing of chess variants broken down by:
- Chess with different starting positions
- Chess with different forces
- Chess with different boards
- Chess with unusual rules
- Chess with incomplete information or elements of chance
- Multimove variants
- Multiplayer variants
- Single player variants
- Chess with unusual (fairy) pieces
- Chess hybrids
- Games inspired by chess
Playing other variants should only have a marginal effect on improvement, and sometimes it's bad(makes you think in ways optimal for the variant, but not for chess). Since others have not mentioned it, playing blindfold chess may help with visualizations, but probably can only be played after a certain level of competency in chess, but I suppose blindfold chess is generously defined as a chess variant. Of course the best way to improve at chess is to play/study chess.
In addition to the two nice answers already provided, I would like to mention the variant King of the hill, whose difference with normal checked is the following: in addition to checkmate, a legal move that moves one's own king to one of the center squares (d4, d5, e4, e5) wins.
This helps a lot for the control of squares. Indeed, one cannot neglect any center square. If one of the center squares is too weak, then soon or later the opponent will take profit of it.