I participated in a flu vaccine study. How do I calculate my expenses for traveling to get the vaccine and follow-up blood draws?

You would have had to track your mileage if this was income as an independent contractor.  Then as you walk through TurboTax Self-Employed, you would need to enter it as vehicle expenses. 

Are you referring to a medical deduction?

Or are you trying to offset your income that you were paid for the study?  It seems quite unlikely your participation of this study was a "business", in which case your expenses would NOT be able to offset any of the income.