Are you cutting the old lawn off, machines can be hired to take off the top layer which can be stacked upside down in a corner and will turn into lovely loam in a year.
The top soil you are getting should be put through a riddle to remove any weed or stones in it. The old lawn once the grass is removed should be dug over and if needed some drainage mixed in. ( for a top class lawn the soil should be dug out, drainage put in then the soil returned) time and money allowing. If not mix some washed sand in the top soil you bought and spread it evenly, rake it level using a plank on edge to make sure there are no dips or humps. The important bit is the gardeners shuffle as you work your way up and down pressing the soil firm then another gentle rake.
Now is it seed of which there are many kinds depending on what kind of lawn you want, or rolls which come in varying qualities, the best money allows.
Seed will show quite quickly and let it grow on before trying to cut it treating any bare patches with more seed raked in. If rolls then make sure all the edges are tight, trim of the spare and then water on a daily basis for at least two weeks.
It is hard work, no easy way and not a quick fix, money is often the deciding factor seed being cheaper than good rolls, that is up to you.
This is a very brief summary of the work ahead, then we are gardeners and know nothing comes easy we have to work at it.