Total gardening novice here.
But... I would like to make some sort of gardening area to grow a few easy veg, partly for eating but mostly for DS (2) to get involved in and learn about gardening. We don't do much messy stuff at home and at least this could be productive and educational - and outside 
We have a decent-sized back garden which is all grass, and could dig some of it up, but the grass is very poor quality (even after various treatments) and I suspect the soil is rubbish (new build, very lumpy lawn so suspect lots of rubble/rubbish underneath).
I have no gardening knowledge or experience. I think I grew cress on a flannel once as a small child, and own a few cacti that I water every few months religiously. I have a limited budget (because I have no idea of costs and don't want to spend loads in case we give up!) and not a huge amount of time to spend on this project - we both work fulltime so a few hours on evenings/weekends only.
A few people have suggested raised beds or planters instead of digging up the garden; others have said to save that cash and just dig up a corner and see how we get on.
So please help! What should I do - dig up lawn or get planters, what will I need to buy/borrow (we own a rake and a lawnmower - no other equipment but some nice neighbours who would probably lend us a spade), what sort of budget should I be looking at, and what sort of time investment?
Oh, and there are a fair few neighbourhood cats (inc our own). Am I just creating an outside litter tray?