It sounds you're dumping the clutch too quickly. You need to find the "medium point" - where you're allowing the clutch to slide JUST ENOUGH while applying power through the throttle. Doing this in 1st gear will prevent stalls and spinouts. It'll make 2nd a lot smoother, too.
Should I be a bit slower off the clutch?
I have the clutch fully in up until I'm halfway through the turn
Like others have said when you get used to driving a manual be careful when driving an auto because your left foot will want to stomp on the clutch and let me tell you it hurts the neck. This along will pissing off the driver behind you.