Excitable Boy has given you lots of useful info. Buying a few plants rather than growing everything from seed this year will get you off to a good start (our Sainsbury's has started doing some veg plants at £1, I think).You could buy courgettes, cues, strawberries, tomatoes, lettuces, brassicas. French beans germinate very quickly and there's still time to start them from seed. Carrots, beets, rocket salad, can still be sown now too. Many herbs will spread from a single plant so they don't need to be grown from seed, except annuals like basil and parsley. Beware of rosemary and sage, which eventually become straggly bushes taking up quite a bit of space.
Yes, plant across the bed, so you can reach to the middle of a row from either side of the bed without interfering with the rows on either side. As you don't need to walk between rows, the rows can be closer than on open ground in some instances, but peas and beans tend to get fungal diseases if they're crowded.
If you keep certain groups of plants in defined areas, this will help next year so that you can set up some sort of crop rotation - make a chart to remind yourself what grows where, as you won't remember next year!
You can plan it all out on paper as you know the dimensions of your beds. You won't be able to grow everything, so stick to crops you enjoy eating, or ones that are never cheap to buy. And of course some things ( like carrots) always taste better than anything in the shops.