Fab - here you go! As promised a list of seed types to try:
Courgettes - F1 Defender (F1 is a hybrid which means you can't save seed and get the same breed next year, but IME all good courgette types are F1s). Plant in either a growbag (2 plants to a bag) or in a pot about the size of a bucket. 2 plants will give you 3-4 courgettes a week for weeks and weeks and weeks! More if you are lucky. Remember to feed and water loads if you are growing in pots (I use a homemade brew from nettles, but tomato food will do, or liquid seaweed mix)
Tomatoes - try Alicante or Gardeners Delight - both easy (you can normally buy plants in garden centres if you want), and produce reliable copious yummy fruit, again for months. 2 plants to a growbag or one to a pot as above. Remember to stake them as they grow (string tied to fence and round base of plant will do). Also Suncherry is a fab cherry type. I always grow about 3-4 of each and make enough sauce to last half the year from anything I don't use fresh (cook in batches and freeze, easier than jars). Germinate in sterile yoghurt pots (use sterile compost) and plant out when big enough (about the size of your longest finger), and remember to water and feed as above. You can also save seeds of the first two types (never bother with the cherry types but I guess it would work if not F1 hybrids) - dry on a paper towel and use again next year.
Pumpkin - go for large types such as Jack of All Trades, and treat like a courgette. The plants grow up things and across ground as well, so a bit of space needed. And very easy to save seeds.
Cucumber - Masterpiece - grow as for a courgette.
Sweetcorn - go for an early type (all F1s unfortunately) - like Early Xtra Sweet. More chance of them ripening in the British summer. You need to grow a few in touching distance of each other, I grow in batches of 6, one per pot (bucket sized). 1-2 cobs per plant, but if you pick and cook within an hour, I guarantee you will be amazed!
Potatoes are easy as well - you can grow Jersey Royals in a bin (look for International Kidney seed potatoes, they are the same) - quarter fill with compost, pop 4 in, top with a bit more, another 4, bit more compost, 4 more and then a bit more compost. Feed and water well as the shoots grow, adding more compost until the bin is full as the shoots grow more and more. Once the foliage has started dying down, take potatoes out as you need them. Seriously yum.
I buy seeds online - thompson and morgan are good, as are dobies.
That should get you going! Let me know how you get on!