Can someone tell me how I would go about wiring an EE ferrite core (see attached) and how the pins work.

I need 7 turns on one side and 35 on another

 

You wrap both windings around the center post. It is best to us a bobbin so you don't damage the wire's insulation on the edges of the ferrite. If you need isolation between the two windings, it is often good to use a split bobbin so the two windings don't touch each other and it increases the creepage distance for safety.

The link below is a good starting place.
http://www.mag-inc.com/design/design-guides/Transformer-Design-with-Magnetics-Ferrite-Cores

 

So how do you connect the wire ends to the pins?

 

Strip/remove the insulation and wrap/solder to the selected pin?
Max.