At my booking appointment (i was about 9 weeks) the midwife did all the questions on medical history for me and other half, height, weight, blood pressure. She also tested my wee to see if there was any infection as this is more common in pregnancy and they tend to do this at each appointment so don't go just before!! Midwife also took bloods to test for blood type, rhesus factor, rubella inmunity, iron levels and also if you consent they test for HIV, syphilis and hepatitis B.
However i didn't get the reulsts until my next appointment with midwife at 16 weeks. Although mine were all fine so if you were really low on iron for example maybe they'd give you a call? Mention to your midwife though and she might make sure she gets that test result back to you sooner.
Tests for downs syndrome vary, in my area you have a test on your 12 week scan and bloods the same day, some people have bloods taken around 16 weeks.
My 12 week scan i had to book myself 1 week after my booking appointment once notes would ahve arrived at hospital after booking appointment. Some poeple's midwives or GP's make the appointments for you and you get a letter in the post. Just ask how they do it all where you are as it does vary in different areas.