What should I check next to speed up a slow PostgreSQL multi-table join?

[Jerry.W]

I'm a junior database administrator working with a PostgreSQL database that's starting to slow down significantly as our dataset grows into the tens of millions of rows. A particular reporting query that joins several large tables is taking over a minute to run. I've added basic indexes on the foreign keys, but I'm not sure what the next steps are for effective SQL performance tuning. Should I be looking at query execution plans, considering partial or composite indexes, or is there something fundamental about our table structure or the query itself that I should investigate first? What are the most common bottlenecks you've encountered in similar situations?