I wouldn't worry too much about him acting out because of something at daycare - I'm a stay at home mom (and my son has never been hit) and my son who is 12.5 months is doing the same thing - he hits when he gets really excited. His hitting is not aggressive - but still hurts! I think right now its a developmental thing and I agree with the other posters who suggested redirection and continued patience as the best course of action. My husband and I hold our son's hand and show him "gentle touches" on our faces and arms and also show him how mommy and daddy touch each other gently. He's really into imitating right now so when he sees us doing something, he tries really hard to recreate our actions. We also praise him whenever he uses gentle touches.
As for saying no, we also have the tantrums - again I think it is developmental. When he is really kicking and screaming, I tell him I understand he is frustrated because he can't do something but that it is my job to stop him from (whatever unsafe thing he is doing) until he is able to stop himself. If it is a big tantrum, I either hold him until he calms down or if that isn't what he needs, I put him in a safe place like his crib or pack and play and let him calm himself down while I take a little breather - I usually stay where he can see me and he normally calms down within a minute. In public I generally just ignore the behavior - my trick is to pretend that he is someone elses kid - that way I don't get frustrated with him or myself or embarrassed and as a result I'm able to keep an even tone when I talk to him. Usually his public tantrums occur if we are leaving some where he is enjoying or if he sees something he wants to play with and is told that he can't. I just keep calm and explain to him that its time to go, or the toy/ball/doggy doesn't belong to us and has to stay at the store/park/friend's house. In these instances, it seems to help to have him wave bye-bye to the object of his affection. I don't know why, mabye it helps him feel more in control? Anyway, it might be worth a try...
I also recommend reading T. Berry Brazelton's Touchpoints - his approach is behavioral based and goes through developmental stages - it has really helped me understand where my little guy is coming from and has given me some great tools for how to deal with him when everything seems too overwhelming.
Sorry for the long post, just know that you are not alone and I'm certain that your son will grow out of it! Good luck!