How to diagnose PostgreSQL slowdowns during peak load?

[Mia99]

I'm a backend developer working on a SaaS application where our PostgreSQL database has started to buckle under load during peak business hours, with query times for certain customer analytics reports jumping from milliseconds to several seconds. I've identified some problematic queries and added basic indexes, but I feel like I'm just applying bandaids without a systematic approach to database performance tuning. For experienced database administrators, what is your methodology for diagnosing and resolving these kinds of systemic slowdowns? Should I focus more on query optimization, revisiting our indexing strategy, investigating hardware or configuration parameters like shared buffers, or is it time to consider architectural changes like read replicas or partitioning? What monitoring tools and key metrics give you the earliest warning signs of impending performance degradation?