Go organic and mix the lime into the fertilizer before application. I use pellet form of lawn lime now but used to use pelletized limestone. The soil PH management will help control weeds as most broad-leaf weeds thrive at lower PH than most lawn grasses. As you may know, Lime raises the PH of the soil. Also, over-seeding or starting lawn from seed using endophytic tall fescues make the growing and maintenance of lawns a lot easier. First these grass varieties typically set roots very deep. The specific varieties I use and have been using exclusively for over 20 years set roots up to 4'-6' deep. That meant in the Chicago area with the droughts we've had in recent years I've only watered my lawn maybe once or in extreme conditions, twice /year. Second the endophyte, a beneficial fungus carried via spore in the seed, grows on the roots of the grass plants and produces and injects a plant eater toxin into the plants which can kill any grubs or other insects that try to eat the grass. No insecticide is ever needed! These grasses are also extremely well suited to low "N" value fertilizers as they require less of it and the deep roots in combination with lower nitrogen fertilizers encourage growth of the beneficial organisms that can supply plant useful nitrogen compounds from the nitrogen in the air passing into the soil via the additional porosity of the organically managed soil. High nitrogen value fertilizers can actually kill off those valuable beneficial organisms. One thing though, the endophyte is only viable for the first growing season after harvest of the seed. After that the seed will produce healthy grass but minus the protection and enhanced growth the endophyte provides. Other than PH, I prevent most weeds by application of a corn gluten product that stops seed sprouts from setting the roots they need to survive. See the linked article for more on the value of corn gluten for lawns.
Bill's Will: Misc Musings of a Techie: CORN GLUTEN: MAGIC STUFF?