It's actually SQL Server Management Studio Express. Do I need the full version? How do I connect?
SQL Management Studio Express will work just fine. The normal setup pattern would be something like this:
This connection should work on any home subscriber internet package. Your ISP may be blocking those ports, so keep that in mind.
Ensure your Windows firewall is configured to allow SQL Management Studio access to outbound connections. Ensure your home router firewall is allowing traffic out. If NAT is doing its job properly, you shouldn't need to configure your hardware firewall.
Above all, check your host's knowledge base. They may have an FAQ page that has the appropriate steps documented.
You will need to ensure the SQL Server is configured to allow remote connections.
By default this option is not enabled. You didn't mention which version of SQL, so I can't give specific instructions.
You will also need to be running the SQL Server Browser service, which listens for incoming requests for SQL resources and provides info about instances running on the computer.
Firewalls may also cause an issue, but I'd check the above first.