A child will only portray behavior that they're allowed to get away with. So, you need to figure out how you're allowing it in the first place, for example, when she throws them are you trying to get her to stop? Telling her if she stops she'll get a reward? Asking her what's wrong? Trying to pick her up? All of these things are a reaction from you, which is exactly what the purpose of a tantrum is in the first place. The best way to handle a tantrum is when she first starts doing it, pick her up, not lovingly, but instead pick her up from behind under her armpits, and place her in her room. Shut the door behind you and stand there in case she tries to come out. Keep the doorknob tight so she can't come out. When you do this, do not say ANYTHING to her. The first time you hear a break in the tantrum, open the door and say "oh good you're done, you can come out now." If she starts crying again, close the door again. The first few times you do this, it's going to go on for a WHILE because she has to relearn where her tantrums will get her. She will assume you still have a breaking point, so she's going to try her hardest to find it.
AS far as a store tantrum goes, when my kids were 1 and 2, it seemed like one of them was always wanting to get down when we were at stores. So, I took them in their double stroller into a mall during the day when it wasn't too crazy busy. Everytime one of them started to whine because they wanted down, I would find the nearest hallway like the ones where a bathroom is, most of the time I stayed in a hall where there was a bathroom, and I would go down it away from people and sit down on a bench. I would look at them and say "no crying in stores, when you're done we'll walk". I would pull out a book and pretend I was reading it while they cried. The second they were done, I would excitedly say "oh good, you're done! We can walk now." The first day of this, I was making it about 20 feet before I had to turn around and go back to the bench, but it was okay because it's expected that a child is going to test you again and again. They have to be able to accurately predict what's going to happen in order for them to make a choice to stop a behavior. That's why consistency is such a huge deal. Anyway, after that first day, the second day at the mall was much better. I would start to head towards the hallway and they'd stop before I got to the bench. Eventually, all I had to do was stop the stroller and say "no crying in stores" and they'd stop. Once they had it down, I went grocery shopping. As soon as one of them would whine, I'd stop the cart and say "No crying in stores. When you're done I'll keep walking" They did just fine. Sure, they needed reminded sometimes by doing this, but they got the hint and after that they were totally fine at stores. Even now, they're 5 and 6. Sometimes they start getting a little too excited and I warn them that if they don't settle down they'll have to sit down for a timeout. If they get noisy again, I find a place in the store where people walk by, like in the middle of the aisles where there are displays, and make them sit there for time out. I've even gone so far as making them hold a piece of paper that says "I'm in time out" That way they REALLY don't want to get a timeout in a store.
So, basically, figure out ONE way for each behavior that handles it the best and STICK WITH IT. She'll stop when she sees that you mean business.