On 'regular' blood tests, ie a routine non specific screen, what is usually requested are Full blood count, liver function, kidney function, cholesterol, glucose(diabetes), thyroid function and prostate usually from men over 50. If you want a HIV test you have to request it and it takes a little longer to come back from the lab. If you do have 'illness' something will show up on your full blood count for example high white cells will show infection.
If you want an additional blood test, you should mention it before the needle comes out!!