So, I'm in charge of my group's Twitter account. What should I be doing to make it help us grow?

It’s a 13-person group. We throw gaming (PC and console) events every month.

For the last year, the events have been hosted in residences. We’re moving out into professional spaces that we’ve rented. And now an effort is being put into trying to grow, get our team into other events, and get our name known locally.

Most of my responsibilities within the group are community-based. So, one of the roles is to find a way to use Twitter to our advantage.

I avidly use Twitter for personal reasons, but figuring out Twitter for an organization is giving me trouble.
We only throw one event per month, so it’s not like there’s a big need to post reminding people about a bunch of events coming up. I feel like just posting every few days for a countdown to the event seems dumb. (But I’m the kind of person who, on a personal level, puts everything on a calendar so I always know when things are happening)

So I guess the issue boils down to..
1) What could we as a group do to get people following and interacting with us on Twitter? (I’m not looking for mindless hordes of spam followers, but actual interested users)
2) Aside from using it to just mention when our next event is, what kind of things would be good ideas to announce to people?