Bad first month in my PhD I started my PhD in physics approximately a month ago. Now I am working on my first project with my advisor and a postdoc that follows me. I made several big mistakes for a little task that my advisor gave me, and now I feel really bad. They were stupid big mistakes that put me in a bad light, as if I didn't understand anything. It seems that I really didn't understand concepts. 

I have the bad habit of always waiting for instructions, without taking initiative because in my past studies, professors usually told me what to do next as little exercises. I also have the bad habit of doing things mindlessly and not reviewing them, often resulting in errors.

I know that I am capable of completing this PhD and that I have the possibilities of doing very very well. I have the mental resources to do it(hardware) but the problem are my bad habits(software).

Now I am facing the problem that it will be harder for me to build trust with my advisor. What steps could I take to start building it again? Does a bad start affect things in the long run(like if my first two month are bad, but then I finish great my first year and do well on the other years, will my references for a possible postdoc be bad)?

I am slowly reading articles and concentrating on my little tasks, trying to understand everything I do. Apart from that I don't know what else to do.

Edit: Thank you very much! I will take all your advice and go ahead! Good luck also for the other PhDs
It's your first month. You will be in the program for 3-5 years. Everyone knows this, and you will have the opportunity to redeem yourself. Be patient. Do better in the future. Everything will work out alright.

What's more important is that you understand your weaknesses. Work on them. Become a better person and scientist by figuring out how you can address your weaknesses by either addressing them directly, or working around them.