I have a video I took on my mobile ('cell-phone' to those who don't use the UK term).
I started off in portrait mode and then, while still videoing, I turned it to landscape mode.

What I'd like help with is, I would like the section where I turn my mobile from portrait to landscape to stay orientated during the 'turn'.
The next bit I'm having trouble with is the size of the video after I've turned my mobile, it is taller, so, how do I resize it?

Any help is greatly appreciated.

If you want the landscape part to be fullscreen, make sure that the project settings have the same resolution as your clip in landscape format. Then the portrait mode part will be quite small, but that is how you shot it.

To keep orientation while turning:
Apply a keyframable rotate effect to the clip and set the keyframes at the beginning of the turn and at the end appropriately. If there are glitches you can try to cure them by using more intermediate keyframes.

I've managed to sort out the rotation problem, I knew what to use, just not exactly how to use it, thanks.

The next bit, setting the portrait clip to landscape settings, I can't find where to do that!

Assuming you have a portrait clip with 1080x1920 pixel you can zoom in (position and zoom effect) up to a width of 1080 pixels without quality loss. You will lose a lot of the picture vertically though but if you are lucky the important stuff is in the picture.

You have no chance to get the portrait clip full size and without quality loss into your landscape video, unless you choose a project size smaller than the source material. You did not record enough pixels horizontally to do that. Game over.

As written above you can zoom into a full hd clip up to 1080 pixels horizontally and that make almost a full screen in a 1270x720 pixel project. You will have to zoom out while rotating the clip. If that helps, this is your best solution. Rendering in good a good quality bitrate will produce very well scalable videos that still look fine in full hd.