Aerate the lawn to help the grass grow more strongly then in a couple of weeks apply a lawn weed and feed. That should get rid of most of the dandelions and other weeds. Any remaining dandelions could be dug out (including the root), if you can't get them all out at least try to pick the flower heads and dispose of them before they set seed - hopefully it's not too big a lawn that you've got there.

After that you could re-seed any bare patches with a lawn seed mix.

you can get a long handled weeder with a vee at the bottom if any reappear you can dig out odd ones as you see them

Not sure they make these any more, but you used to be able to get a Weed Pencil (there was a push up wax one, like a deodorant, but I know that's not around any more) which you push into the middle of the dandelion growth to kill it. Backbreaking if there's lots, but pretty effective - there are certainly paint on versions of glyphosphate which will do the trick too, but make sure the plants are growing strongly, the soil is moist, and no rain will fall for a number of hours if you use these things.