If it wasn't for Mrs A I'd leave the flowering weeds becuase I think they are pretty and they certainly are good for bees. Mrs A enjoys being on her hands and knees rooting away with with a long tool in her hand, but I hate it.
If you really don't want the weeds, then a selective weedkiller designed for lawns is probably the easiest. Remember, though, that weeds will grow again from immigrant seeds when they get the chance; it's not as though you can dig them all up and you'll be free for life.