After construction, very little of the original topsoil will be left. Dirt is Dirt. Doesn't matter what kind you have, you do have to know what kind and what proportion (clay, silt, sand...organic matter) you DO have. Soil is basically tiny little rocks and you need to understand what their size means...Then you have to learn to manage whatever you have...right now you probably have a lot of subsoil with no organic matter (but few weed seeds!) Lawns can be grown in poor soils, as long as you know how to improve them over time.
Hopefully, the contractors brought in topsoil with organic matter before they sodded your lawn. The front is what 'sells'...the back, is 'less important'...sigh. Hopefully, the planting beds are mounded higher than the lawn and they installed edges by trenching around the edge of the lawn. Hopefully, if you have automatic irrigation that the two areas (plant beds and lawn) are on different zones.
If not, healthy plants HAVE TO HAVE HEALTHY SOIL. The only way to improve ANY soil type is decomposed organic matter. Bark, wood chips or other non-decomposed organic matter take time and lots of nitrogen to break down where soil organisms are then able to use it for food. Micro and macro-organisms (those you can't see without a hand-lens or microscope (bacteria, nematodes, mychorrhysae...I've got to learn how to spell that word... and those you can see easily such as millipedes, earth worms) need DECOMPOSED organic matter that they ingest, go back down into your soil and poop out the organic debris responsible for water-retention, nutrient retention and aeration necessary for 'good' soil. Try to find a place that 'makes' mulch from human poo and sawdust...and is regularly tested. This stuff is reliable, you know what is in it and what is not in it...no weed seeds, no pesticide residue, a little high in heavy metals from us humans taking prescriptions (so don't use in your vegetable gardens, sigh), beautiful, fine-textured, dark taupe in color. You will notice a marked change in your plants after mulching with 'this stuff'! Careful with fertilizing in conjunction mulching your beds with this stuff...in my area it was called "Gro-Co".
When your lawn is established next year, aerate your lawn pulling plugs and leaving the plugs to disintegrate, dump little piles of decomposed organic mulch all over your lawn, rake into the grass, top seed patchy areas (remember to KEEP MOIST until new grass plants can be mowed). Repeat every other year.
Tilling is not good if you have a high percentage of clay...what is it that makes concrete? Lime, gypsum, clay, gravel, water and mixing. If your plant beds are not done, I would use a spade and fluff up my soil by double digging...once. Soil organisms will do the work of mixing organic matter into your soils. All you have to do is put the 'food' on top of your soil.
For now, using an organic lawn fertilizer with bacteria and michorrhyzae innoculants should produce a very healthy lawn. Organic fertilizers take time versus the POW! you get from inorganic fertilizers. That POW! in my opinion does not make for healthy plants. Once you've begun mowing (about 1 week after sod, 2 weeks after seeding...or allow to wait until the second mowing if too sparse...) water deeply and only when the soil dries out to encourage your grass plants to grow deep roots. Great way to eliminate weeds and shallow rooted weed grasses. Use a line-trimmer to keep the grass from going any farther than the 2-4" trench defining the edge of your lawn.
Test the pH of your soil...lawns need a higher pH in general than most plantings in your beds. Make sure that you get a professional test and add lime if below 6.5. If your soil has a pH of 6.0 for instance, I would do half the recommended amount of lime to raise the pH to 6.5 by the instructions and test again a month or two later. I would then do a second application using what I learned about the results of the first application.
If you have moss, don't sweat it! That is a good indication of where there is too much moisture, no competition or too much shade. Moss is a great opportunist, where there are bare areas and/or too much moisture, a pH that is more acid you will find moss. It is not a serious competitor of grasses. Moss control uses sulfur and causes the pH to lower. Moss won't grow if there are no bare spots, lots of sun and you allow your soil to dry out before the grass needs moisture again.
If you have similar grass mix to those I am familiar in the Pacific Northwest, make sure you don't mow any lower than 3". I've seen the difference between 2", 2 1/2" and 3". Huge. Our grasses in the NW have huge root systems and need photosynthetic top growth to support the roots. Once in balance you have a VIGOROUSLY growing plant that can shade out, out-compete and thrive even when the top few inches of soil are dry.
Make sure your mower blades are SHARP. If not, you will see a dustiness on your very green, healthy grass where there is necrosis from 'ripping' by dull blades.
A simple test I've used to tell if my lawn is dry enough to water, is to step on your grass and the blades don't bounce back leaving a footprint. Use a spade and make sure the water gets down to 4-6"...you want to be able to use 1" per week of water eventually. That is the standard. Depending on your location, humidity, rainfall, type of grasses...this might be entirely different for your needs.
Mowing is important to lush lawns. A minimum of once per week. Twice a week is better. At 3" height the grasses grow at a slower rate, allowing once per week to be enough for that 'just mowed look' and possibly be able to not bag your clippings. The proper organic fertilizer will have bacteria added that are responsible for decomposing thatch! Organic fertilizers don't have to be applied as often and the 'wow' factor is amazing. Amazing deep green healthy grass over a period of weeks. Don't ever do synthetic fertilizers with broad-leaf herbicide, in MY opinion. You shouldn't need any herbicide if you water deeply, infrequently, mow on high and feed your soil. If you do, use spot herbicide. And make sure you use the right fertilizers (definitely organic) for the fall application! Too much nitrogen at this time can cause major problems with fungus...
Lastly, if you have moles, voles, making piles of soil on your lawn...let them be. They eat (mostly) grubs in the soil, aerate and top dress your lawn for free. Just knock the hills down and rake into the soil. Lower your 'expectations' just a bit and you won't need to control everything. Anything that advertises fast results, complete kill...stay away from.
Sorry if TMI! I am working on this problem of mine...grin.