I’d recommend installing a Pleroma server. It speaks ActivityPub and you can use any of the nice Mastodon apps with it. I've run a Mastodon server for the last 9 years, and wouldn't recommend Pleroma over it for a large many-user instance, but it's relatively tiny and lightweight for a personal server. You can configure it not to talk to the rest of the Fediverse so that it remains your friendly, isolated silo.
Pleroma looks to be very twitter-y. I don’t feel twitter is a great model for a small tight-knit group. For a larger less familial group, it’s probably better suited.
Like, i’m thinking photo album sharing (twitter-like makes photos ephemeral, quickly disappearing on the timeline) and conversation (twitter threading has never been strong imo).