In Sept 2016, I got a letter from IRS that I own them taxes for 2014. I replied to the letter and after some back and forth, this month (August 2017), I finally got a final bill, which I paid in full to the IRS. IRS already received the check and applied to my bank account.

Now, my question is, WHAT WILL HAPPEN WITH RESPECT TO STATE? How do I pay state? Should I wait for the state to contact me or should I contact them? What is the normal process?

#1;You are supposed to pay the money owed for income taxes at the time you file your statetaxes. If you can't afford to pay them you have a problem. Most states charge you a late payment penalty and all of them charge interest on the debt. There are several ways you can handle the situation you get the best chance for a deal if you initiate it. Waiting for a taxing body to contact you about its money is never the way to go. You don't have to pay your state taxes in a lump sum, taxing authorities will work with a taxpayer who is willing to pay and is just having financial trouble. It is always best to pay as much as you can at the time you file, even if it's only $5. It shows you are willing to pay and you acknowledge the debt. Your state will work out a payment plan with you if you contact them. Frequently these can be on reasonable terms depending on how you choose to pay.
#2; as mentiopned prviously.