Find out before your users do

My app is live at <url>. Stack: <stack>. I currently have no
monitoring.

Set up the minimum that's genuinely worth having:
- An uptime check, and what URL it should hit. Not the homepage —
  an endpoint that actually exercises the database.
- Error reporting that captures exceptions with stack traces and
  sends them somewhere I'll see.
- An alert channel that reaches me, and a rule so one outage isn't
  fifty messages.

Recommend specific services with free tiers, and tell me what each
step actually costs me in time.

Then: what should the health endpoint check, and what should it
deliberately not check?