I am not sure if our first date went good or bad. But I'll give some details.

I am 24 years old she is 25. After I talked with her for a few days, she called me on the phone and I asked her out. She told me that for her I am childish because of the way I talk.

For example the question I asked her "How would you imagine your first date?", feels too her childish she told me. But I don't think it's childish, i think it's good too know what kind of thinks she likes.

Now let's talk about the date. I took her from her apartament with my car, she said that she has a female friend that drives better than me, and also told me that I drive like a beginner. That is not true, but I tried to act like I am not affected, but I am affected because I now that I am not a bad driver. Also she says that I am childish because I don't have a job yet, but soon I will have because I finished univeristy.
She told me that she is very mature for her age and stuff like that, bla bla, and she was a little bit touchy with me which makes me think that she is attracted to me. Also she says that my "childish side" can be solved in time + getting a job, which makes me think that she wants a relationship with me.
After the date we still talked on the phone she was open, friendly, but I am not sure what to do next. I asked her for a date, and she says that she is busy this days, I said "No problem" she replied that we can still go out when she will have time because this week-end she is going home in her town.
So what should I do next? Should I text or call(show my interest) or wait for her too call me more often(not too appear too needy). Also how can I establish the alpha frame over her, because I don't feel so comfortable about her opinions over me (I now that I am not childish, and not a beginner driver).

If you are thinking this then 95% chance that it went badly. She already said she was "busy" these other days. Cut her out cold, if she becomes unbusy let her contact you. You need to focus on volume until your game skill level catches up, then move on to quality. In the meantime dont pursue problem/lukewarm chicks as you are going to waste 90% of your effort on 10% of your results.

finally this whole "childish" issue is a complete red herring, its the wrong answer to the wrong question, forget about that.

Thank you for your answer! I think that you are right. I will cut her out and see if she is interested.