Be reassured that the effects are quite limited and will resolve without any care in a couple of weeks.
A bruise develops where the needle was inserted. This is much less likely to happen if you press over the site with cotton wool for several minutes with your arm left straight (not bent).
As with any wound, an infection may develop where the needle was inserted. See your doctor if the wound site becomes red and inflamed painful and increasing over time
Rarely, some people feel faint during a blood test. Tell the person doing the test if you feel faint, as you should immediately lie down to prevent fainting
If it is growing, contact your doctor. If it is not growing, has no redness beyond the bruise and is not painful, you may leave it alone and it will resolve. It may go through a change of colors like brown, greenish-blue and yellow