I suggest looking around at a number of casinos before making any decisions. Especially look at the deposit requirements, minimum bets etc. There is quite a variety of Internet Casinos out there. Many will let you visit for free and even place some bets without signing up (You won't win any money, but you can play for free).
Once you pick a casino that you like, you will be asked to sign up. Usually this will mean filling out an online form. Many will not even require that you give your name and address. You will end up with an account number and a password. (Remember, if you only have an account number and password on file with them, and you forget them, you have lost your money).
The next step is an important one - depositing money into your casino account. The next section details this process.