How to list all the indexes along with their type (BTREE, BRIN, HASH etc...)
The types of indexes in Postgresql are stored in the pg_am catalog table. So to get a list of all tables and their indexes with the index type (or access method ("am") as PostgreSQL calls it) you can run the following

SELECT tab.relname, cls.relname, am.amname
FROM pg_index idx 
JOIN pg_class cls ON cls.oid=idx.indexrelid
JOIN pg_class tab ON tab.oid=idx.indrelid
JOIN pg_am am ON am.oid=cls.relam;