Mastodon new user info
You can opt-out of search engine indexing and hide your follows & followers by going to preferences (the gear at top left) > other.
You can also set your posts to automatically delete after some amount of time using this handy tool: https://forget.codl.fr/about/
All that said, nowhere is truly private online so be wise about what you share. Keep in mind, too, that one of many flaws with the decentralized fediverse is other instances may not respect your instance's settings.
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!