You mentioned not knowing anything about Azure, so checking out the tutorials at www.windowsazure.com is a great place to start, especially if you're attempting to launch a production application into an environment you know little about.
Having said that: If your SQL Server is running in a VM, and that VM is part of a different deployment from your asp.net app (e.g. you used something like a web role, or Azure Web Sites), than you can not use your SQL Server's internal IP address, since you'd have no access to that IP address (each deployment lives in its own private network). Your options would be:
- If you deployed your website to Azure Web Sites, you'd reach your SQL Server VM through its public endpoint (you'd need to create that endpoint, called an input endpoint, with port 1433 opened). So it would be yoursqldeployment.cloudapp.net:1433.
- If you deployed to a Cloud Service with a web role (yourwebapp.cloudapp.net), you could do the same thing as with Web Sites, or you could create a Virtual Network between your Cloud Service and your SQL Server Virtual Machine deployment.
- If you deployed your website to a virtual machine in the same deployment as your SQL Server VM (and it doesn't sound like you did that), then you can directly communicate with SQL Server via its internal IP address or DNS name.
And, as long as everything is in the same data center, you won't have any data traffic between your website and your database going outside the data center.
One more thing, regarding Azure SDK: While you may need the SDK for accessing storage and other services, you don't need the SDK for communicating with another virtual machine.