psql: fatal database does not exist error
psql assumes that you want to connect to a database, you can either provide one (just after the command) or it will assume you want to connect to a database whose name is the same as your username (or the account name of the process that started psql):

So, you could write:

psql my_db


or

psql -d my_db


or yet

psql --dbname=my_db


... and psql will connect to the (local) database named my_db.

You have more connection options that you can check in the documentation.

If you don't have yet a database created, check createdb.