I'd start off with things like salad leaves (which are expensive in shops and in my house at least half a bag is always wasted) and herbs (ditto above). Potatoes are easy to grow and I do grow a couple of new potatoes but they do take up space and are relatively cheap to buy in shops. Ditto broccoli etc - I do grow them because I love gardening but they take a long time to grow and use up acres of space.
I also tend to grow veg which are hard or quite expensive to buy in shops eg. baby beetroot, baby carrots, small, sweet, peas, small broad beans, blueberries in pots etc.
In fact I'd start off by working out why you want to grow your own, is it for eg. to get a huge supply of organic carrots or is it (like me) because you're a bit disorganised about planning meals and it's a huge relief to know that at any time you can go outside and pick a great salad or put together a decent stirfry with spring onions, courgettes, carrots etc. (have made myself hungry now)