I never learned phlebotomy in nursing school
but was not so good at starting IVs, so I was terrified when I took a job on an acute floor where the RNs did their own blood draws. After a couple of tries (and practicing on my husband), much to my surprise, phlebotomy has turned out to be one of my better skills.
Don't tie the tourniquet too tight. This can create too much pressure in the vein and make it easier to "blow." Sometimes using a blood pressure cuff is better than the tourniquet.
Don't go for veins you can see, go for the ones you can feel. Ask friends and family members to let you "feel them up." Tie on a tourniquet and practice feeling for veins.
Don't go in at too high of an angle or too deep. You may be going right through the vein, especially if you're leaving a lot of bruises.
Use a butterfly until you're comfortable with the skill set. The flash lets you see the "pop" you should feel as you enter the vein. Once you see the flash, don't go any deeper.
Practice, practice, practice. Admit to the people in your office that you are not feeling very confident and ask if you can practice on them.