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.
I like them steamed: Steam , unpeeled, for around 35 minutes depending on size. Use paper towel to rub off the peel after it's cooked, and then slice.

They're very likely to stain your nice towels, clothing, and counter-tops .... a cutting board is recommended.