The first step is to open an account at an online casino. If you don't know any online casinos, you could try Casino-On-Net:

Opening an account is similar at all online casinos, but we will look at it from the point of view of Casino-On-Net. First of all, at Casino-On-Net, you should ignore the section on poker. Instead look at the left hand half of the screen. If you click on "Quick Play" you will have the chance to try out the casino without downloading anything. If you click on "Play Casino" the casino software will be downloaded onto your computer.
Next you will have to provide some information about yourself. You will need to provide accurate information; if you later deposit money using a credit card, this will be used to validate your card. If you give wrong information, you may find it difficult to make deposits.
If you downloaded the game, you will now be able to choose real money or pretend money. For now, choose pretend money, and we will give some of the games a try without risking anything.
Click "roulette". If you downloaded the software, you will be offered a choice of a private table, a public table or a reserved table. Choose a private table for now; you don't want to be distracted by other people when you are getting used to playing online. You will then be asked what betting limits you want. With pretend money, these make little difference, so you might as well choose the lowest limits.
You will now see some piles of chips in the bottom right corner. Click on the value of chip that you want to play with, then click on the roulette table to place your bet. You can place all the standard roulette bets. When you are satisfied with the money you have staked, click on "Spin" to spin the wheel.
The other games work in a very similar way. Now you have played roulette successfully, you should find the others fairly straightforward.