Covering indexes are a really careful thing to investigate, sometimes you'll get big wins out of it but often times those indexes can easily be misaligned with really minor application changes so it's something I tend to approach in very limited situations.

Vacuum & Analyze are two fun ones and I think postgres in particular would do well by investing a bit more into the default configuration of autovacuum since the out of the box settings are pretty conservative compared to the state most out of the box DBs are used - but I think that DB configuration is very much an unsolved problem in general and can be a quite surprising source of major performance gains (or losses!).

I mentioned mat views elsewhere - but if you are really hitting quite predictable usage patterns (where covering indexes are stable enough) then some well defined mat views might go a long way to tuning performance that little bit more - just be aware of the fact that rematerialization isn't instantaneous and if you have a high rate of change in the data then you'll probably want time based re-materialization instead of trigger based which means you'll need to account for some inconsistent data (as is the case with caching in general).