I am a newbie using mySQL on unix (Mac OS/X). I have done simple operations, such as CREATE DATABASE, CREATE TABLE, SELECT, etc, but suddenly I find that I cannot do update operations, such as CREATE DATABASE.

Here is my results.
mysql> show databases;
+--------------------+
| Database |
+--------------------+
| information_schema |
+--------------------+

mysql> create database ladsondb;
ERROR 1044 (42000): Access denied for user ''@'localhost' to database 'ladsondb'

mysql> grant ALL on *.* to richardladson@'localhost';
ERROR 1045 (28000): Access denied for user ''@'localhost' (using password: NO)

I suspect this is a basic problem, but it is a show-stopper for me.

Yup looks like a permissions problem to me
if you GRANTed permisions on the command line client, did you FLUSH PRIVILEGES afterwards. Mind you I'd also want to check the syntax of the grant statement you used

An alternative way of setting permissions is to use MySQL Adminsitrator which is part of the MySQL GUI tools from MySQL.

mind you I'd also want to check the syntax of the grant statement you used