When you add a failover group to an Azure SQL Server, you'll define the Primary and Secondary servers and then be able to configure which databases are included/excluded from the failover group.
Once configured correctly, you will be able to see the details in the Azure Portal (SQL Servers > > Failover Groups blade). On the Configuration details tab the primary and secondary servers will be displayed and their statuses. Additionally, 2 endpoints are listed. One for read/write (the primary) and one for read only (the secondary). You'll almost certainly require the read/write endpoint. It'll look something like:
You can then use this endpoint in your connection string and let azure handle any failovers. You'll always connect to the Primary.
Essentially you can then swap out the failover group endpoint for what used to be the server name.
A basic connection string example:
<add name="core" connectionString="Data Source=<failovergroup>,1433;Initial Catalog=<databasename>;User Id=<user>;Password=<password>;" />
Once you have your connection strings set up and working you can test the connectivity to both regions using the Failover fuctionality in the Azure portal on the failover group blade for the server. It's usually pretty seamless.
There is an msdn article with tips & tricks for using failover groups, should you have any other issues relating to the actual logins. But without further info on the nature of the connection issue it's hard to recommend if it's appropriate to your situation.