As a server rookie and Parse user, I need to migrate and I intend to migrate to Parse Server, likely with Heroku and mLab.
Once I have clicked Migrate and Finalise in the Parse Dashboard, all data from my original Parse client code goes to the new database, right?
Once migrated, I can just push an update of my client code with the new Parse Server SDK pointing to the new server?
My main over ruling question is do I need to do any management on the client side, sending data to both servers? Or does Parse migration handle this?
I think you are mixing two different things. Read the tutorial
Simply
Step 1 You should move your data from Parse.com to self hosted database (mLab or mongoDB and more...), this step means that api.parse.com will use the "external" database but you will still use the code and server from Parse.com (when you send query to your app it goes to api.parse.com and than it access the database) - do this till end of April 2016
Step2 move from api.parse.com to your own instance of Parse server (the one you download from github or install it on heroku). You will need change you code in your app because it wont use api.parse.com fro mthis point... - till end of July 2016
On github the developers still say that it is not "production ready". You should only migrate your database and build the whole server later. You can read the discussion here