Question: I've just pulled some beetroot out of my garden.

I've never cooked beetroot before so would love a few suggestions.

Answer: 
Basically you boil them or roast them until tender, then let them cool and peel them. It is better not to cut or peel them before cooking, they will bleed lots of color and flavor. Some classic things to do with them: puree into a soup called borscht, slice them in a salad (particularly nice with goat cheese), serve as a roasted vegetable like you would a potato or turnip. They especially love dairy products with a little tang such as sour cream, feta cheese, yogurt and so forth.
My Russian Flatmate inspired me to try 'beetroot chocolate cake'. The beetroot keeps the cake really moist but without the need to use much butter/oil. It's really healthy and tastes very rich and moist, almost similar to chocolate browny. I made two of them in the last couple of weeks and will make another one tomorrow.
As per recipe, I tend to change mine over time, but as a rough guideline, you can use a carrot cake recipe and replace the carrots with beetroot. And of course add some chocolate :). My cakes both had about 1/2 kg of beetroot in it.