It will be MANY, MANY years before you can reason with your children. She is at a great age to start a time-out or cool down or quiet time. One minute...One and a half....tops. After, get on her level, look her in the eye and give an explanation by a hug. Keep the language simple..."no hitting" "no biting" etc. Now that my daughter is three, I make her tell me why she was put in time out before the hugs and kisses.
A great way to deal with tantrums at this age is to head them off by giving her choices. Do you want to brush your teeth or get on your pajamas first? Do you want this diaper or this one? It doesn't matter so much that they are the EXACT same diaper...they just like to feel some control. Red shirt, green shirt. It takes a little practice but after a week or so, it will become very natural for you to automatically offer a choice. It doesn't work for every situation...and sometimes a choice doesn't exist...but it certainly can help with the gazillion thinkgs a day that irritate a toddler.
As for the food, my daughter has always mostly eaten what I eat. She was easily two before she had anything resembling a chicken nugget. I was pretty easy-going about food and let her have whatever she wanted...BUT I only had healthy choices in the house. If she wanted to eat yogurt (no HFCS) three times a day, she got it. That is a real-life example...she ate Greek yogurt all day every day for weeks. Any time she gets on one of these jags, I find it best not to fight it. It will pass.
I suggest you try some choice with food as well and be casual about other foods. Just make sure you can offer her choices that support her overall nutrition over time.
I got the best advice from my pediatrician...don't worry so much what she eats in short period of 1-2-3 days...but what is she getting over a week or month. If you can balance intake over time and introduce new flavors, you'll be ahead in the long run.