M.P.
Separation anxiety is high at this age. And it's really painful for the parent to leave for both the parent and the child. I've been thru the clinging, crying situation. However it does get better as your child adjusts. He's afraid that you won't be back. He'll learn thru repetition that you will and he will gain confidence with that knowledge.
You can help this adjustment by doing just what you are doing. Leave even tho he is clinging and crying. Assure him that you will be back. Show him that you are confident that you both can handle this and quickly leave. Don't hover or hesitate. He will stop crying once you're gone. Return in 30 minutes; then in an hour; lengthening the time that you are gone. Go somewhere. Just going into a different room doesn't work because he knows you are there.
THis is difficult. And he still may cling and cry for several weeks especially if he senses that you are hesitant about leaving. He will pick up on your emotions. I've found it very difficult to leave when my daughter and now my grandchildren cling to me. But we have all become better at leave taking over time. My 7 yo granddaughter still asks for one more kiss, one more hug, etc. It went on seemingly forever until I said "one more hug and kiss and I'm gone" and left without responding except to say "see you later" with a smile on my face. No more prolonged good byes. I started out with this attitude with my grandson and for a year or so he would cry briefly when I left. Around the age of 3 he began just running off to do something else. I think his mother taught him to do that because she would steer him into an activity as I was closing the door.
I was the one to take her when my granddaughter started pre-school at 3. The staff encouraged me to stay for awhile. Before long she was happily playing with the other children. At her request and the staff's encouragment I returned to share lunch with her a couple of times. I don't think she ever cried when I left tho she did sometimes cling and beg me to stay. I gave her a hug and a kiss and said, "I'll see you later." If a child didn't run up to her asking her to play a staff member got her involved in play.
Perhaps this worked because of her age and/or her temprament. But it would be worth a try if you would be more comfortable making it a slower transition.
I've also heard recommended that the parent and child visit the day care a couple of times before the child actually has to stay when the parent leaves. Spend an hour or so with your son exploring the center and then leave with him.
The goal is for him to be comfortable with his new surroundings and to know that you will return to get him. I think at 2 he still doesn't know that someone still exists when he can't see them. Therefore it is important for him to feel secure in day care. The workers can do alot towards developing that security. Be sure that you feel comfortable with them.
I empathize with your situation. It will get better.