INDIAN WELLS, Calif.—Teenager Daria Kasatkina has been on a roll, getting to the third round of two straight majors and breaking into the Top 50. This season, the 18-year-old Russian has scored wins against Venus Williams and Timea Bacsinszky, and a doubles victory over Martina Hingis and Sania Mirza with Elena Vesnina. She reached the quarterfinals of Indian Wells before losing to Karolina Pliskova on Thursday. In this Q&A, she talks about her big tennis dreams, her small-town personality and why she likes to watch Rafael Nadal.

You've been doing very well recently. Were you expecting to have so much success at a young age?

I never expect something, because if you expect something, always it's bad. So I just go on court. We had very good preparation … If I am in the quarters, I deserve it.

You said you want to get into the Top 20 this season. Do you feel you're on track to do that?

Yeah. Now especially, why not? But I am not focused on the ranking. I am focused on my game, and I want to improve my game every day, every match, every practice.

I changed [my] coach [in] October [2014], and now my results [are] better. We make [a] very good job for one year. I improve my game, my ranking. I think a lot of good things [are] in front of me.

What are your career goals?

My dreams are very big. I want to win [the] Golden Slam. Why not? But … it's too far. But it's my dream.

It's never enough. So I will go, and go for my dream.

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How did you start playing tennis?

Because of my brother. He used to play. He was playing just for fun. And he said to my parents, ‘I have to start to play tennis.’ I was six years old. For two years, I used to just play two or three times a week. And I start to play some tournaments, I start to show some results. And we start from Russian tournaments, Tennis Europe, ITF and now WTA.

Who are some of the people who have been the biggest influences in your career? The most important?

My brother, [Alexsandr], for sure. He's supporting me from nine years old. He's my fitness coach. For sure my parents, they are with me. My coach, Vladimir Platenik. For one year we are working together. My fitness coach from Slovakia, Maros Molnar, and the place where I'm practicing, EMPIRE Tennis Academy. They help me a lot, so that's all the people who support me.

Who are some of the players who have inspired you?

Difficult to say, because I don't look for other people, other players. I just focus on my game and trying to improve every day and every practice.

I like to watch men's tennis. I like to watch Rafael Nadal because he's my favorite player. I don't like to watch women's tennis, because it's always strange ... because they are playing not so good like men's, and it's better to watch men's.

For sure, serve and moving. That's two things they're much better [at] than girls. And they're more disciplined, really, than us.

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Some players from Russia choose to go elsewhere to train, and some players choose to stay in Russia. Are you still based in Russia?

I'm practicing in Slovakia already for one whole year, and I like it. Because in Russia it's very difficult. If you want to improve, you have to go [to] Moscow, but I don't like Moscow. That's why we go to another country. And it's better to practice [in] Europe, because everything is [around there]. For me it was better to go to Europe.

You said you don't like Moscow. How is your home base different from Moscow?

Moscow is [a] very big city, and I don't like it. [There are] a lot of people, a lot of traffic, and I hate it.

[I’m in a] small city, it's called Trnava. Just 75,000 living there. Bratislava, if you need shopping or you need fun, you just drive 20 minutes and it's there.

Speaking of shopping and fun, what do you like to do off the court?

I like to spend time with my friends, if it's possible. I like to watch football—I support the football club Barcelona, because of my brother, again. And I like to play football. I like all sports. And sometimes I like to lie on the bed and spend some time with my phone.

What are some of the best experiences you've had as a tennis player?

OK, my first best experience was [at the] U.S. Open. I was [a] lucky loser [and] reach[ed the] third round. From this moment, I start to believe I can beat good players ... I play very good [at the] Kremlin Cup, [reach the] semifinals from [qualifying] and we won doubles with Elena Vesnina. Those were two very important moments for me because I start to get confidence, and it helps me.