Anya Taylor-Joy played the part fairly well. You could track the evolution of her character from smart-naïve-innocent-yet-defiant young girl to intelligent-yet-still-defiant young woman. She didn't let the era define her i.e. chess being a boys club and the idea that girls should join a social club, get pregnant, and start a family. I also like that she didn't let anything stand in her way. Of course, she got cocky and suffered for it, but she appeared to have learned her lesson and she was able to achieve her goal but more on that later.
As for Beth's relationship with her birth mother, I think it was fairly obvious what happened in the first episode when her mother's finals words were "Close your eyes." When the reveal of the actual events happened in the final episode I wasn't surprised. On the topic of mothers, I'm glad they expanded on Beth's relationship with her step mother and I like how their relationship grew with their openness and honesty with each other. They were both lost souls that found and had each other. I was genuinely sad when she died.
As for Beth's father figures, I really liked the janitor character and I would have liked more backstory about him. I was hoping the series would revisit him but unfortunately it wasn't until after his death that the story went in that direction. As for her birth father, they kept showing him, and I thought he may pop up or, at some point later, Beth would seek him out. Her birth mother's suicide aside, I think the big reveal was that her father actually moved on and started a whole new family. Sadly that event was the final nail in the coffin from her mother's POV. So in retrospect, I'm glad neither sought out the other.
Plot-wise, the arc of season 1 felt fairly straightforward. While it was mostly predictable, I still found it enjoyable.
The only thing I didn't like is that the ending didn't feel earned. Beth had spiraled down to the point where she was burning all her bridges with her friends, yet everyone came to her aid in the end to help her - why? I would have liked to see Beth have a heart to heart with Beltik (Harry Melling) and apologize before going to Russia. Then he could have been the one to get everyone together to help her. It didn't make sense to me that Watts (Thomas Brodie-Sangster) did it without some outside catalyst to influence him. It also didn't make sense that everyone traveled to New York when most of the people that helped were from/living in Kentucky. Maybe Watts felt bad that he didn't go to Russia with Beth and this was his attempt to make it up to her. I don't think he owed her anything, again, especially after the way she treated him.
Regardless, I think the heart of the entire story can be found in her conversation with the young Russian boy. After you achieve the rank of grandmaster, what happens next? What do you do with the rest of your life? I think if a season 2 is made it could explore that question and personally, I would be interested on how Beth would answer those questions.