You have to go application by appication. It all depends on the amount of data and downtime allownce you have.
For some applications it may be feasible to bring down the application, move data to netapp and start the application from the netapp. You could do a backup/restore. Simplest but large downtime.
For other cases you can have both storages visible to the server and setup a robocopy /mr to slowly copy data to the netapp. Then bring down the application, do a final robocopy /mr and bring up the application from the netapp. This could give a smaller downtime.
For databases in general you have large files and you could use logical or application replication from inside your application.
There is software doing this, but in general it is expensive. This could give a smaller downtime.
You can use hardware solutions, like SAN "replicators/redirectors", much more expensive, but gives the smallest downtime. There was a company called Reldata doing this, then bought by Starboard, not sure what they are doing now.
NetApp used to have an excellent software for this, called Topio, first OEM, then bought the company in 2006, in 2007 redesigned and enhanced it as ReplicatorX (http://www.netapp.com/us/company/news/news_rel_20070405.html) and then suddendly discontinuing it shortly being announced, probably because of the then crisis and cuts. The excuse was that there was not a considerable market for it!