You have to stick by your guns with kids this age. You say no, and you stick with it. Use time out, and don't EVER give in. You give in one time, and you're going to pay for it. Everyone who deals with your daughter, has to do the same thing. If she is disobedient, you give her one warning. If she doesn't listen to your warning, you put her in "time out". This can be a corner or a naughty chair. She has to sit there for three minutes. (if she does this all the way through) you go and talk to her. Talk to her like an adult. Ask her if she knows what she did wrong. If she doesn't then you explain to her what she did wrong. Then you let her get up. However, the very first time you do this, she is going to TEST you. She's going to get up, scream, cry, kick, etc. Once her butt leaves the chair, you get up, DO NOT LOOK AT OR TALK TO HER, pick her up and put her back in her time out chair. From this time, the 3 minute time limit starts over. First week or more of doing this, it's going to be he!!. But after that, she's going to start understanding what's expected.
Small kids cry a lot just to seek attention. If your daughter talks to you and you do not pay attention, then she will cry and throw out thing. If she is disobedient, find a how you can discipline her. Lock her in her room and let her cry there for some time, talk to her and tell her the mistake she has. Give her an assurance that every time she does a mistake she will go through the same process.
We had the same problem during the threes and early fours. We started giving him "time out" in the corner. We chose a corner because no matter where we went. A store, a restaurant, grandma's house, etc. You can always find a corner. This did two things...1) redirected his thoughts so he wasn't screaming while in the corner or trying to get out, and 2) reinforced something educational. When he was finished, we would get down to his eye level, and explain why we were not happy with his behavior and make him apologize to whomever he had misbehaved to. It worked for us. All kids are different, but it worked for us.
Trust me though, be prepared for the stares and comments from others. I just ignored those people. My theory was that if I allowed my child to misbehave because I was afraid others would say something, then I was teaching him that it was okay to act up which would only make things worse both at home and in public.
My kids are 24 & 22 parenting is one of the hardest jobs you will Ever encounter. There are no second chances you must get it right the first time. At 3 years they are testing you and it only gets worse from there. Maybe talk to your pediatrician?