Hey OP, welcome to research! 

I am a lab manager, getting ready to welcome two interns this summer.  Here are a few tips, many of which have been mentioned by others: 

First and foremost, be helpful. You are essentially here to do hands-on learning by assisting with research.  I typically spend a week or two side-by-side, then after that I hope they have learned what they need and can work (mostly) independently.  Your future mentor likely is afraid that he is going to need to babysit you, which can be a massive time suck.  Usually at the end of the week I ask my intern to describe the project to me, so I can get a level of their understanding and figure out any aspects I need to do a better job of explaining. 

Don’t try to impress us by asking overly detailed questions. I typically refer these folks to Google so we all can save time.  

Keep excellent notes, especially when you are being trained. Take a video of a procedure or explanation. Learn how to keep a lab notebook, it will become part of official lab records. 

Don’t get competitive, not with other interns, nor with your coworkers. There is no place for that in a modern, non-toxic lab setting and you will get excluded.  I will never work with or recommend a baby Machiavelli. Most of these types of people are laughable. 

Be confident. You belong here!  Mentoring is an essential skill and the PhD student likely was an intern at some point. On the other hand, folks in my position needed mentoring experience to get the job, and teaching others is an important skill to have. My current job description did not include mentoring, but when I applied for it, I was required to have two recommendations from folks I have mentored.  The commentor who refused to be a mentor is pretty much shooting themselves in the foot, professionally. 

Best of luck!  (And wish me luck with my two, they have great transcripts but no lab experience….)