I had the same/multiple scenarios when my daughter was that age. She is now 3.5. It drove me crazy, but I stuck with it. I kept taking her with me..if she didn't behave, I would leave or sometimes I stayed if it was something I really needed to do.
I noticed that if she thought she pulled me into her tantrum, it would go on and on. Once I picked up on that, I ignored her as long as she wasn't hurting herself or someone else. Just don't let him control you..because that happened to me. You stand firm on what you need to do and eventually it will change, but it will take some time.
Also, I noticed my daughter would have more tantrums if she was tired/hungry. So, I kept snacks in my purse and only tried to schedule certain things after a nap. Sometimes it was neither of the two and she still wanted to do it her way. He's just at this age..when being independent is very important to them..just set the ground rules before you leave...while you're in the car...I just kept reminding my little one about how I expected her to behave.
It will get better..just hang in there and TAKE A STAND! You're the grown up.
I thought I would never be able to give advice to another mother b/c of what I experience with my daughter. I thought I was going to lose it.
I got stares from many people...but hey I'm the parent and she wasn't being hurt and I didn't care if she screamed while I shopped b/c she had to learn she was not in control.
GL!