You normally have a blood test at your booking appt and again at 28 weeks. The booking one is a few different samples but only one needle.
The blood tests look at blood group, iron levels, infection screens for things like HIV, hepititis, etc. The 28 week one rechecks your iron level.
Depending on what test for Downs your hospital does you'd either have another blood test at your dating scan or at 16 weeks, if you wanted a Downs screening.
They're your basic tests, depending on how risk you are or become for other things during your pregnancy it may be recomended that you have further tests.
If you're one of the 15% of people who are Rhs neg blood group then you'd be advised to have an injection at 32 weeks, and also a blood test and possible further injection post delivery.
If you have any risk factors for diabetes a blood test would be advised at about 24 weeks.
I do sympathise, I used to be very phobic about needles and blood tests. I remember crying at midwife appts. I think it did actually help me get used to it, I'm a blood donor now so I must be better!