Totally normal! Definitely look into happiest toddler on the block for now, then parenting with love and logic for later.
In the meantime, start sounding like a broken record and say "screaming is not allowed" and "hitting is not allowed" and change her location (In love and logic they say "If you can't change the behavior, change the location"). At home, she goes into a playpen or crib or into her room. Give her time to get quiet, or 5 minutes, whichever comes first (this is my own personal rule, you can figure out what works for your child, but at this age 5 minutes is enough)
And be ready to do a lot of baby hauling. If you're at home and this happens, say your line "___ is not allowed" and haul her into the room/crib/playpen. If you're in a restaurant/mall whatever, fine a place. In a restaurant you can go to your car. In the mall or grocery store you can find a corner or the restroom (yes, we've stood in corners at Kroger before -- pushed the shopping cart over there and stood with my back turned!).
You just need to be prepared to be inconvenienced a few times. This is the main reason we don't follow through with punishment/threats, is because it's inconvenient to enforce it and kids figure that out quickly. They know you won't actually DO what you say, so they don't bother minding.
As far as screaming in the car, I've started bringing a book or magazine in the car on outings. If there are fits and screaming in the car, I pull off the road, turn the engine off (except on hot days) and just read my book. The iPod works great too because you can put your earplugs in as a visible cue that you're not listening. If it's really bad, I get out of the car and stand next to it (only had to do that once).
Now, I just have to start looking for a place to pull over and they straighten up. I still have to pull over (have to follow through, even though it's inconvenient!) but we only sit there a minute.