First time to implement logging.
We do not have the problems with how (technically) to log. The problem is logical - what to log.
Any general recommendations here, or it is purely case-specific?
Generally people, get their logs in one place, realize they are missing something, add it it, realize it's noisy or expensive, turn some things off. It's generally iterative and what you need for your dev/biz ops / customer support. You'll find more challenging questions about who / when can access logs than deciding what to log. Do make sure to get audit logs sooner than later, though your cloud provider (we use GCP primarily) may be doing this for part of your infra
Checkout Google's SRE book, there are a few relevant essays, especially as you look to alert on log events.