The first thing you need to do to start using Google Apps is sign up for it. You can sign up for a Standard Google Apps account at this URL. There are also Premier and Education Google Apps editions, but we are going to focus only on the Standard Edition as it is the most widely used.
On the sign-up page you will see two options for registration depending on whether you are the administrator of the domain (you own or manage it) or a member of it (you are a regular user who cannot manage the domain).
We'll assume you are the administrator of your domain and choose the first option. Then type your domain name in the empty field and click Get Started.
On the next page you will be asked to fill in your details (name, email address, etc). Fill in the required fields and place a check next to "I understand that if I cannot alter DNS records for my domain, I may impact my organization's ability to use Google Apps".
If you wish, you can fill in some details about your organization as well. When you are ready, click Continue to proceed.
Now you should create your administrator account. Type a username and a password in the corresponding fields.
Make sure you read the Terms and Conditions for using the Google Apps service. When you are ready, click I accept. Continue with set up >>.
This completes the Google Apps registration. You will be logged in to your Google Apps Dashboard, from where you can activate and manage your Google Apps services.