Normal toddler behaviour?

I have a 3 month old baby and a two year old toddler.
The two year old has always been a difficult child and anyone that has spent any length of time is quick to admit that she's hard work.
I love her with all my heart but constantly feel like she hates me and that there's something "wrong" with her.
Long before I had baby she began only wanting her dad and then after an unplanned c-section and new bubs needs she went to me less and less. She constantly screams and grunts at people. Family or strangers.
She goes to childcare two days a week and for the past few months has been crying walking in saying "no kindy". She's been going since 5 months with no previous problems. Today the teacher said "She's a funny little thing she screams if someone touches her and other times she's so good we wonder where lillli went" :(
She screams for dad in the car, she runs from me at the shops, she had major melt downs when she wants a cuddle but I'm breast feeding.
I rarely get to, but I sometimes have a day with just her and she's such a good and happy girl but all other times she's a nightmare.
I don't really know what I'm asking but does anyone else feel totally unloved and unappreciated? In so sad at how she sees me. I do everything for her yet she still loves her dad so much more and feels like she hates me.