You can use your State Department of Revenue's website to check status of state tax refund. Choose your state here How do I track my state refund? and it will give you steps to track the status.
If you are not able to find your return in their system or it shows no record; that means they have not started processing your return yet. Until they start processing/reviewing it, their system will have no record of it.
And processing times vary from state to state. Some states will put a notice on their refund tracking page to let you know how long processing takes; some do not do this. If there is no notice, just be patient and keep tracking on their site. They'll update the status once they have processed your return and approved your refund.