We had been TTC for almost six years before our positive test, so we were quite surprised that our GP just went with us reporting a positive test and didn't do any tests to confirm the pregnancy. (We didn't really believe it ourselves!)
I was offered an early scan at 10 weeks because of my underactive thyroid, because it apparently increases the risk of miscarriage and they want to check that the pregnancy is viable. So we had a 10 week scan and a booking in appointment, followed by the regular 12 week scan. There's also a big gap between the 12 week scan and the 20 week one. Now at 31 weeks its been a while since I last saw my baby, but there is movement to reassure me.
It does feel odd to find out you are pregnant and for nothing to happen for a while, but its quite normal. Early scans are usually offered if there is particular reason to suspect miscarriage or an ectopic pregnancy, which could be symptoms in the present pregnancy, a maternal history of miscarriage or ectopic pregnancy, or a condition like mine which increases the chances.
You can get a private scan if it would make you feel better, but as others have said, getting scanned won't prevent something sad from happening, it just lets you know if it has already happened. :(
Congratulations on your pregnancy, all the best.