What is the process of changing Reserve units?

YOU NEED TO CONTACT YOUR COMMANDER NOW NOW NOW
I see Soldiers in your situation all the time.. who do not get the correct information, then just move.. and think its all honky dory. NO
YOU MUST get a DA FORM 4651 signed by your GAINING UNIT commander BEFOORE YOU MOVE. Then you must have your current unit's commander sign it BEFORE YOU MOVE.. DO NOT MOVE WITHOUT TALKING TO YOUR COMMANDER AND GETTING THE DA4651 SIGNED.
Who is your Unit Administrator. I'm curious why you would have to ask this question here before going to them. Do you have a UA or an RPAC? Who is your Squad Leader? Who is your Platoon Sergeant? Who is your 1SG Sergeant. All of those people should have been contacted!

Ok, you need to get with your command now. Like today and talk to the person that is Active Reserve in your unit so that they can get the paperwork started NOW and get you in at the very next drill to speak with your commander. There are some forms that need to be filled out but that can be done between the two units, and all you need to do is sign them. Should be easy.

You need to do EXACTLY what everyone is saying here. Don't assume that there's an open slot either. I tried pulling this off my way amidst units getting deployed to Afghanistan. I should have asked what I needed to do waaaay before making plans in my life. You have the same mindset as I did. Why would you think it would be so easy? I can tell you that often it's not. Because the Armys needs come first and slots can take forever. Not trying to be a jerk but think before you make huge life decisions. I was immature and did a lot of dumb stuff hence PFC.