What can kill Weeds and Moss

If your lawn has a lot of moss, it means you have a lot of damp shade. I would forget planting grass as it will be less than satisfactory. I would consider some other ground cover or a dense planning of shade loving plants like hostas. I have an area like that that I planted with hostas, ferns, heuchera, astilbes, lily of the valley, liriope, and a number of other shade tolerant plants.

Since you think that a lawn is worth pursuing then you have a couple of options. You could kill off everything with Round-up or remove what's there with a sod cutter and then prep the area. Then you can seed the area, cover with straw, and keep it watered. Laying sod is another option. Depending on where you live, starting a new law may be better in the fall than the spring.