If I was going anywhere w/ the kids when they were younger (now 9 & 5) I first made sure they weren't hungry/tired, etc.
Then I would set expectations (based upon age) before we would enter the store/house/park/??.
Today we are going shopping so we can have some food in the house for us to eat. When we go in the store I expect you to ...
In our case if the kids wanted to use the child shopping cart they had to stay in the same aisle, not run, no running the cart into others (especially mom!) and to come immediately when I ask.
The expectations are then followed with the consequences (logical ones)...
If you can't follow the rules for using the cart...then I will give you one warning, if you still can't follow the rules then we will return the cart and you can sit in the big cart until I'm done shopping.
AND FOLLOW THROUGH!!
You of course will have different expectations and consequences based upon where you are going and other circumstances, but if you have realistic expectations (ie not sitting still in church for 1:45 hrs) with logical consequences and use both consistently, you can achieve a change in behavior. It won't always be perfect...but everyone is happier when we know what is expected of us, little ones just need to be reminded more often!
FWIW, I only had to return the child's cart once for each kid...after that they always followed the cart rules (and can still recited them now that they are older!

).
I can also look back and see that the more challenging outings were the ones where I didn't set the expectations. In the beginning I had to be a bit more formal about it so I would remember, now after 9+yrs... I do it almost automatically.
ETA... my kids weren't truly screamers/tantrum throwers...but the one time I had to return the cart for ds for not listening he lost it. I put him in the regular shopping cart and finished up my shopping. I reminded him that he had his chance and didn't listen and that he knew what the consequence was. I was almost done w/ my shopping and he only cried for a few minutes so it wasn't horrible, but if he had continued I would have just taken him back to the car until he calmed down.
Tantrums are always about getting a reaction from YOU... the more calm and less reactionary you are, the quicker the tantrum goes away. If you find it to continue to be an issue then I would just consistently remove your dd from the situation until she calms down.
It also may be helpful to be aware of what is triggering the outbursts. At 2yo I found redirecting useful. Instead of saying don't touch that and nothing else, it was more like, we don't touch XXX, oh look lets go play w/ the truck, etc. You are still giving the message, but following it with something that distracts them from what they originally wanted.
Unfortunately, when they hit 3/4yo they are single minded about what they want and are less likely to be distracted...but then they are older and understand the rules better. I found 3/4 to be a bit more difficult as we transitioned through to 5yo and a more even understanding of what they can/cannot do.
Good Luck and I hope this all helps in some way?