Adirondack Mountain News posted a very good summary of what happens in New York whenever a hiker needs help from the rangers.  This is specifically the process that is done in NY where we're relatively unique in that the Rangers are in charge of all SAR activities on state lands.  Other locations in the country it could be any number of agencies and isn't always consistent with who is in charge.  That said though, the overall process and the insights into how the searches are run should be consistent across the country, if not world.  The full post by them is here:

https://www.facebook.com/share/p/1X2E9aYc6c/

There are a few take aways that you should get from this article.

1. Leave a plan with a responsible adult.  Tell them where you're going, who you're going with, when you're starting, when you should be out, when you'll check in, when to start to worry and what to do if you reach that worry time (call 911, or the DEC Forest Rangers directly at 1-833-NYS-RANGERS (1-833-697-7264).  Both will get you sent to the ranger's dispatcher.)
2. Sign in at all trail registers.  Include your phone number and where you are going.  These registers are only checked in the event of a SAR event.  The rangers will call those that signed in and gather intel on the missing person, if they were seen, where they were seen, how they seemed to be faring, etc.  Doing this can really help the rangers find the hiker, but it can also help you if you get lost or need help, they will know where you have been.
3. Be prepared to spend the night.  This means have the 10 essentials with you, and make sure they are sufficient for the weather the day of the hike, and for the weather the day or two after.  Rescues do not happen in minutes, they take hours to days, even with aerial support.  The satellite SOS devices are great, and they will greatly reduce the area and time that the rangers need to search for you, but they too are not instant.  Expect them to take 30 min to an hour before a Ranger is actually notified and dispatched.  The Rangers tell us to expect them to take 2 hours for every mile you're in from the closest trailhead, and that is if they know exactly where you are.