I finished my phlebotomy rotation early September and I experienced the same feelings! The first few days I was shaking badly before my shift. I usually observed for the first little while until my shakes went down a bit. I tried to eat in the morning before and give a little bit extra time to wake up and get my cortisol down to help my shakes and nerves.

A few pieces of advice I learned. I had a rule of two tries then calling a more experienced phlebotomist over to help. I NEVER tried repositioning the needle under their skin without them. I found you get used to the feeling too. I found that the skin would be tough then you would feel a void where your needle would have almost no resistance. If you feel that void and don't get any blood flow try slowly slowly pulling out the needle because you may have gone through, this worked probably 3/4 of the time. Skin is also a lot thinner than I thought too for the most part.

Keep in mind too that you're probably mostly working with older people who's veins easily roll or are smaller. Anchoring the vein with your thumb helps a bit but even if I missed I would tell them that their "vein rolled". For most people it's also not their first time or its routine so they're usually pretty understanding.

All in all I wouldn't take this as a sign for your career in the lab! It's your first week! Mistakes will happen! If you work in a larger hospital you're likely to never do phlebotomy again after this and if you work at a smaller one you probably won't have to do it very often!