Allotment. ...

So happy for you getting your first allotment I know how happy and eager I was when I first got mine which I no longer have.
The answer to your quest is yes carboard will do the trick providing you cover the carboard with soil on top the carboard will keep the weeds at bay until the temperature it creates kills the weeds.

Agree with 3d, its so exciting! Do take photos periodically so you can see how much you've achieved (and show us!) And you are lucky to have a shed. Neither of mine had one and the second one was just a field that we were moved to when they wanted to build on the other one. And they never did build on it either...

But, if you cover the whole plot with (hopefully) well rotted manure you won't need the cardboard. The winter rains will soak the nutrients from the manure into the soil if you leave it open and the birds will have the change of removing any soil pests that venture near the surface.
Also depends on the soil texture to some extent. If its heavy clay and lumpy leaving it open to winter weather could help to break it down if we have some good frosts.
Usually the purpose of black plastic is to kill off growing weeds as much as prevent new growth so I would use it on the parts you haven't dug yet. Six months of that treatment will save you a lot of hard weed removal next spring.

Just my experience - others might not agree...good luck anyway and don't work too hard!

Exciting ... Good luck !

I must say growing veg is great fun, but the digging is not.
Keeping down the small weeds that grow in between the veg is also a pain, but eating the veg and watching them grow is wonderful. Keep a diary of times, failures and successes.

You are going to have a great time growing all your veg and fruit and it's equally great to learn that allotments are still around. So many seem to be threatened around here.

Happy digging and planting!