How do I set up high availability at the instance level?
It sounds like Failover Clustering is what you may be looking for.  You can have the instance failover (though there will be a period of disconnect for end-users, but that is typically minimal while the passive node becomes active and SQL Server starts up).  SQL Server Failover Cluster Instances (FCI) are the way to achieve high availability at the instance level.

It is noteworthy, that this technology sits on top of WSFC respectively, and you will require shared storage (this will be different if you are on SQL Server 2012 implementing AlwaysOn Availability Groups, but that is not instance-wide failover).

Failover Cluster Instances are a robust technology within themselves, much more that can be directed through a Stack Exchange answer, so my recommendation is to use the above link and if you have any more specific questions feel free to ask them on a separate question.