Hi, I have bought the attached EE Ferrite core and want to know where to put the turns of wire. Are they wrapped around the middle part of the E?

I need 35 turns on one and 7 on the other.

 

For typical transformer and inductor use, yes.

You wind the wires on the plastic bobbin and then put the core on the bobbin.

If you don't have a bobbin it is a good idea to get one because it makes winding a whole lot easier. If you don't have and can't get a bobbin, wrap the core with tape (any kind) to prevent the enamel (assuming magnet wire) from being scraped off while winding.

 

Does this look right?

 

No. The windings must turn around the center leg of the core. The windings should be in planes parallel to the core gap planes.

 

Like this?

 

Basically yes (1st pic). But you should put the windings onto the bobbin (2nd pic). Both, the 7 turns and the 35 turns. Make sure insulation of the wires is sufficient.
Otherwise the bobbin makes no sense. After the windings are on the bobbin, you put the bobbin with the windings onto the core.

 

Not sure what you mean by putting the windings on to the bobbins. I thought the wire wrapped around the middle leg with the terminal wires coming out the back. then the middle leg would go in the bobbin slot.
Is there a diagram I can see?

 

It should look like this:
http://en.wikipedia.org/wiki/Magnetic_core#/media/File:ER_core_assembly_exploded.png
Maybe you made it right, it is just hard to see from your picture.

 

So when wrapping the 2 windings on the bobbin what is used to stop them shorting?

 

Looking at the ee core they seem to touch inside. Should they be touching inside? All I got with the package was 2 e cores and the plastic bit, do I need the tube shaped bobbin?

 

The wire must have insulation, otherwise as you say, a lot of shorts whereever they touch.

 

The bobbin shape is for the specific core. The one shown in your first picture is right. Tube shaped bobbin is for the cylindrical center leg of the core.
Wind the wire on the bobbin with out the core, and then insert the 2 halves of the core into the bobbin, and terminate the wires on the pins of the bobbin.
Insulated magnet wires provide the insulation between the turns of the winding. After winding the first winding, you can use a tape, and then wind the second winding. The tape provides insulation between the windings. Too much tape turns, then better insulation, but increased leakage inductance. Search for youtube videos on something like 'ferrite transformer winding'.
When the core touches fully, it is ungapped core. Gapped / Ungapped core depends on your application.