Can a Bitcoin miner process their own transactions for free? Can I become a Bitcoin miner by setting up multiple nodes to give higher priority to my own transactions to process them for less fee or for no fee? 

My understanding is that a confirmation is when a miner includes your transaction in a block he calculates and submits. So can you setup like six or more mining machines which gives higher priority to your own transaction if you know the transactions output addresses already? Or is it just an automatic process which you can't change?

If it's possible then how exactly I can achieve this?
A miner certainly can do that, and some miners do (well mining pools actually). Some mining pools like ViaBTC and BTC.com offer services where they will prioritize a transaction for you for a fee (paid out of band). There have also been many cases where a miner will make a transaction, pay a 0 fee, and mine the transaction themselves.


  So can you setup like 6 or more mining machines


The number of mining machines or nodes has nothing to do with this. You could do this with one machine and one node. However in order to actually profitably mine and have a chance of actually finding a block, you will need a large number of ASIC miners, much more than just 6 machines, more like something in the hundreds or even thousands of ASIC miners.