Use compost, pelleted lime and grass seed. Avoid all chemicals. Don't worry too much about a few weeds. Green is good. You don't need every blade of grass to be identical.

There are plenty of pet-safe, kid-safe herbicides out there but they all tend to be nonspecific so will kill or damage any plant they come into contact with. Whether their usage makes sense depends on how many weeds you have.

Deep rooted weeds like dandelions or rurmex (dock) may be best removed manually.

So no fertilizer just the lime will work with the seed?

Whether or not the lime is a good idea depends on where you are, and the pH of your soil.

Liming my soil would be a waste of lime, and quite possibly hurt a plant or three.

For weeds get a hand weed puller and pull them out one by one.

Probably aerate and dethatch the lawn before spreading compost and seed mix. Let the grass get thick and long before its first mowing.