Getting started is easy. Create an account and a user name for yourself. This user name will become the all important @twittername that will be displayed with your Tweets. It is possible to have separate accounts so that you can use something like @JamesSmith for general Tweets but @Astronomer123 when you are Tweeting about a particular topic or interest.
When creating your account you can select from various privacy options under Settings. For example, it may be useful for followers to know where you are based and, if Tweeting from a mobile with GPS enabled, your Tweets will include the town or part of town where you are when posting.
You can choose that only certain people will see your Tweets instead of them being public. This can be useful if you are considering asking a class to use Twitter accounts to share comments or research a current or recent event. Only you and the class, perhaps, would be able to see these Tweets.