In your first year, a wise move would be to keep your spends to a minimum - cardboard will do the job at keeping weeds down just as well as black plastic, and you should be able to find it free - it'll do the job till you get round to digging those parts over.
Certainly worth investing in the plastic if allotmenteering becomes a passion.
Your other question about raised beds, is about how you want to garden - in theory, you don't need to dig over raised beds over every year. Some say digging destroys the soil stucture.
Personally, I like digging (saves me a fortune on gym membership) - and by using scaffold boards for paths, I think it'll give me more flexibility & space for planting throughout the year. So I've chosen not to have raised beds. But I'm sure if you look around your allotment you'll see that lots of people are fans of raised beds. It really is a matter of choice.
As to manuring.... I'm not bothering this year, but I plan to get hold of lots of manure for next year. The plot I've taken over hasn't had anyone really growing anything on it for a number of years - just a series of people that have given it a bit of a go and then given up 'cos they realised they didn't really enjoy it or have the time to commit. So I know the soil should be very fertile.
Not sure what you mean by the soil being one spade deep... I'm guessing you mean that this is how far you've dug it over. If that's so, then if you dig trenches to plant your potatoes, I'd just loosen the soil below with a fork - just to make it a bit easier for the roots to find a way through.