I'm certain the answer is different for every school... and some students with clinical exposure try their hand at phlebotomy before even starting med school.
At Hopkins, you'll formally be taught how to do this (phlebotomy and IV) during the Clinical Skills course in the second year. We practice on each other at first, and are given opportunities to round with the phlebotomists and IV nurses. Remember, see one, do one, teach one. Sounds silly, but this is true for these techniques. If you round with someone, they'll probably do the first case and expect you to do the rest. Many students also run IVs while shadowing anesthesiologists (a popular activity for first and second year students at Hopkins).
Cheers,
doepug