Many lawns are maintained in reasonably good condition for years without any fertilizer treatments, but there are undoubtedly many which would benefit from a little plant food. Often, this is as simple as going to the garden center and buying a good lawn feed, the types which are watered in are best. The granular varieties may need to be watered in or used just before it rains.
Any feed high in nitrogen will make the grass greener and grow much faster which means a lot more mowing? A more balanced feed should used at the end of the growing season. Don't cut it short before the winter or you will get moss. The best and cheapest approach to a weed free lawn is good management. A thick, healthy sward created by regular, but not over keen mowing and suitable fertilizer treatments which will resist invasion and spread of weeds.
Your turf has been fully weed treated prior to delivery, but some species of weed may take a further week or two to die off. Any odd weeds which survive the treatment should be eradicated by spot treatment with a proprietary weed killer. Sprinkling sand into your lawn will encourage the finer grasses and discourage the spread of weeds.