Seed will give a better lawn, but turf can do well. It requires different prep and maintenance.
It can be done cheaply, but you need alot of effort.
The goal is at least 100mm of good quality topsoil, raked till it's fine and not over compacted then very gently and very well leveled.
You generally make the top soil yourself with what you have already got existing - add sand if the existing stuff is heavy or clay like adding loamy topsoil to break it up if needed.
If it's light and sandy then you'll need to go the other way and add some peat - otherwise it will dry out and not support grass growth.
If you have the money and height you can just add bought topsoil untill happy!
When prepping the top soil you can start major leveling if needed, but take care to maintain your depth of topsoil by not moving too much from one place as grass will grow poorly in sub-soil. This is one of the main reasons for growth differences in a lawn or colour variations, where a patch has subsoil in its rootzone.
Doing this in summer creates work. Frosts will help break down the soil. Creating this prep will be hard day's graft with a rotorvator. (If you dig in autumn you can leave it dug out and the temperatures will turn it all grainy and easy to break down).
Once the major leveling is done you can let it dry out a bit, then rake it to a semi firm and very level state. I don't like using a roller or foot method to ramp it down, a big bit 2m of flat wood with some weights on it dragged around works best. Don't compact it too much or you will invalidate all your digging and work. Being level is what makes the difference to the final lawn.
Then a day or so before putting the seed/turf down treat the soil with pre seed fertiliser. Water evenly as required, which is more than you expect. Away you go. Lawn. With future requirements to overseed, rake and feed.