I know that, I need to upgrade the storage engine after upgrading the binaries. But I am not very sure whether I should upgrade the authentication after upgrading the storage engine. Or it does not matter?
As part of the MongoDB 3.0 upgrade it is strongly recommended that you upgrade to the SCRAM-SHA-1 authentication scheme. New users will have their credentials created in this format by default; mixed authentication schemas will likely lead to confusion.
You will need to upgrade your client driver & admin tools (if you're using a GUI) in order to use the WiredTiger storage engine. Drivers/tools that have been updated for MongoDB 3.0 will also support the new authentication method.
As per the note on the SCRAM-SHA-1 documentation (referring to a future major release which would like be 3.2.x):
The next major version of MongoDB is likely to remove all support for MONGODB-CR.