Since I only get called after phlebotomy has tried and given up and nursing isn't taught phlebotomy any more where I am (so the more experienced nurses can be helpful, but the junior ones in general have done less than a medical student) I am in the uneviable position of having to get blood or else!
Practice, practice, practice. It will save you hours of sleep as an intern to be able to put in an IV or draw blood quickly, especially if you have a VA in your residency.
My mental algorithm for phlebotomy looks like this (IVs are a whole different story):
1) Ask pt where they have tried already and how many times they tried (you would be surprised how few times people try before they give up)
2) If pt reports multiple sticks in forearm and hand, I take a quick look at them with a tourniquet on. If it is because their hands are swollen with edema I have them put their hand on their bedside table and lean on it with my hands for a while to squeeze out the edema and then go for it. Otherwise, I skip straight to #4
3) tourniquet above the elbow and look for a visible vein anywhere that looks reasonable and try with a 20g butterfly
4) if no vein: do an anatomic stick of each antecub, based on my knowledge of anatomy, I usually try both the antecubital and the basilic veins, and sometimes the cephalic just to be complete.
5) If that fails (not to often) I got to the foot and take a look, usually there is a visible vessel and I slap some lidocaine jelly on it and come back in about 10 minutes and come back (no study shows this works, but it makes it feel cool and patients convince themselves it doesn't hurt as much, the wonders of the placebo effect). If there is no vessel, I access the vessel in the crook of the ankle (if you dorsiflex the ankle you will feel a little fossa right smack in the middle of the anterior part of the ankle where a branch of the saphenous is). If there isn't a good flow there, I see if I can draw the saph. vein on the leg.
6) If I'm really striking out: radial art stick, if having trouble consider marking the artery using a doppler
7) finallly when all else fails femoral stick (I really don't care if it is the artery or vein to be honest). This is something you can numb them up for since you know it's going to be there.
I'd say 95% of the time I can stop at #4, 4.9% of the time I can stop by #6, and I can count the number of fem sticks I've done purely for venipuncture on one hand probably. I've done the dangle thing (not unusual for me to kneel on the floor with the hand hanging off the bed to draw a smaller hand vein), but not a hot pack, I'll have to try that. Makes sense to me.