I used to have an allotment, but those are hard to get these days. But at home, things like new potatoes can be grown well in potato sacks, main crop (big spuds like King Edwards) need to be in the ground ideally. Spuds tho can be superb yeld and will last absolutely ages if left in soil and in the dark and cool (sunlight makes potatoes go green). I used to still be eating my King Edwards for Xmas dinner even tho they finished growing around Sept.
Courgettes, peppers, toms, lettuce, cucumber, beans (with a climbing frame or poles) onion, beetroots can all be grown very easily in pots, just have to make sure watered and fed enough throughout the season. I tried once, when gave up allotment, to grow corn on the cob in pots. They grew fab, they were awful to eat!
I'd recommend just having a go at couple of your fav's and see how you get on. If enjoy it, try more things next year, it is good fun tho!! Oh, and saving money, vegetable seeds are often cheap this time of year as season ending, as is a lot of gardening things.