I am tasked with organizing chess training for a team (of 6 amateurs, elo around 1300-1900).
Other than playing games against each other, what other activities can we do to help to improve chess?
Doin chess puzzles , studying positional elements , studying endgames ... Lots of things u can do ..
Dvoretzky method: set them up endgame study positions you know and then play it against them in clock simultaneous display.
Replay their losses on a demonstration board and discuss analysis with them all.
The improvement of each player on a team is usually an individual task. If you are hoping for individual improvement, treat it as such.
***If you are organizing the team, my advice would be: don't put pressure on yourself to improve the players. Get them connected to sources of improvement outside of yourself - it is the same problem that is asked constantly on here: how do I improve.
What you can and should do as an organizer of a team is make sure the players are aware of the extra rules that go with team play, assign a team captain that understands what a captain should be doing during the match (assuming it will be a live tournament), and deal with the strategy of scoring during the tournament(accepting draws etc).
The first and absolute least you should do is to establish cooperation, not competition. Too often teams are managed like a survival of the fittest. The "best" member gets all the plaudits while the "worst" member is left to shame. All of this is caused of needless competition. Remember that teams compete not against themselves but against other teams.