Temper tantrums are hard to deal with as a parent when you see your child crying, not being able to communicate and causing a raucous in public
Not much to add to Walter Davis' answer except for a few points:
I doubt you are doing anything wrong. Every kid is different.
1. Ask your daughter what she is feeling/wanting, that might ease things, but sometimes not. Frustration and/or boredom is hard to express in an articulate manner when you are six so they need to know they can tell you what is on their mind and that it's ok to take their time doing so. However, kids can not be entertained all the time nor get what they want.
2. This part is really hard, and it requires being calm and an even neutral tone. If the tantrum doesn't stop, go to a time out spot, usually a corner. Speak to here at her level that it is fine to be mad, angry, and state that they are angry. Also let her know unconditionally that it is not appropriate behavior to have a fit, and that she is in time out in the naughty spot. Let her know that every time she leaves that the clock begins again. Then just let her sit/stay there for each minute of her age (age 6 = 6 minutes). She can do whatever she wants while she is in that corner for the 6 minutes. If she tries to get out, just put her back. If she tries to interact, do not do so. Read a book, sip your tea, etc.
When the 6 minutes are done, reiterate why she was in the corner/naughty spot, and ask her to say sorry. If she doesn't, begin again. If she does say sorry, say that's great, give her a big hug, and tell her you are proud that she said sorry, and that you would love to continue and talk with her.
3. Ask yourself when and where she is doing this. Is it at the grocery store or regular errands? She might be bored, so plan it so that she can be a big girl and help you. Is it at events? Plan to bring activities she might enjoy or games you can make out of the events.
4. Start getting good manners dvd and storybooks that demonstrate good and bad manners. And give her praise when she is being helpful, and give her a reputation that she can live up to (your granny loves it that you say please and thank you and don't shout, you do good deeds all the time!). She will pick up on it.
5. Sad to say this, but if none of the above work after a month or 2 of solid reinforcement, it might be time to say that if she doesn't behave, it is time to leave right away and that you are dissappointed. Then pick her up and go. Do not interact with her during this time.