🛠 Tooling: Set up automated releases with Changeset

[JoshuaKGoldberg]

Bug Report Checklist

• I have tried restarting my IDE and the issue persists.
• I have pulled the latest main branch of the repository.
• I have searched for related issues and found none that matched my issue.

Overview

This is my first monorepo and I've never done automated releases before. I'd like to try out Changesets. 🚀

Additional Info

No response

[JoshuaKGoldberg]

Current status: the Changesets action is running on this monorepo but I haven't fully set it up to actually do releases. I'm still running pnpm build, pnpm build:release, pnpm publish, etc. manually locally. Which is not ideal.

#822 has links to places where I've been manually publishing...