Potatoes are an interesting plant, they can even grow in just moist rotten straw.
For baby potatoes, you can just dig the plant once a week, and cut off the small potatoes then carefully replant the entire plant, spreading out the roots with the too tiny potatoes that are left on, then repeat each week. Naturally you need soft loose soil for this, or try straw.
Potatoes tend to grow near the soil line, so a mulch or loose soil needs to be side-dressed around the plants, any sun will 'green' and ruin the potatoes (even just a little sun can get through a too thin layer of mulch).
For the best tasting potatoes, wait until the tops completely die. Don't fertilize after flowering. Potatoes are related to the tomato, and sometimes you'll see little green "tomatoes" in near the leaves, these can be cut and fried and kinda taste like a tomatoe.
Good Luck,
