Diagnose and optimize slow PostgreSQL queries on massive data beyond indexing

[Zachary76]

I'm a data analyst working with a large PostgreSQL database where some of our core reporting queries have started to time out as our dataset has grown into the hundreds of millions of rows. I've added basic indexes on the columns in the WHERE clause, but the performance gains were minimal, and I suspect the issue is with inefficient joins and the order of operations in my more complex queries. For database specialists, what is your systematic process for diagnosing and optimizing a slow SQL query beyond just indexing? How do you use tools like EXPLAIN ANALYZE to identify the true bottleneck, and what are some advanced optimization techniques you turn to when simple fixes aren't enough, such as query restructuring or materialized views?