A quick shortcut is to keep the edge of the lawn trimmed - even if the lawn is a bit shaggy, and there are weeds in amongst the plants, neatly trimmed edges to the lawn helps to make the garden look cared for.

You can use a glyphosate based weedkiller on the stones. It's the least problematical to animals, but keep the dog and your daughter away from it when you're spraying or the area is still wet. Use it when the weeds are growing fast. They take the glyphosate into their system, and die about 2-3 weeks later. It'll kill the deep rooted weeds that keep re-growing (dandelions and so on). You'll still have regrowths of weeds from seed to deal with.

If you want to grow veg you'll probably have to fence it off from the dog.

Get all the soil planted up as densely as you can. Don't go for the "bright flowers with clear soil between" look - that takes a lot of maintenance. If you can end up with no soil showing a) the weeds will struggle against the competition b) any weeds that get through won't be noticeable.

Talk to neighbours and see if you can get offered free plants - enthusiastic gardeners usually have spare plants that they don't have room for. Try also freecycle/freegle - though it's probably best to wait to see what's offered rather than to put in a request.