Measure the area you want to cover with soil. For instance, 50' x 100' is 5000 square feet. Laying 3 inches of soil over this area would require 1,250 cubic feet or 46.3 cubic yards. HERE is a calculator. The type of material doesn't matter as that only affects the weight

More soil will go into low spots and the rest will provide a level seedbed. How much soil you actually need is a guess because this soil will also settle. Most dump trucks will haul from 10 to 16 cubic yards of material. You can get a larger dump truck, but remember you have to have an empty spot to dump whatever they bring. Find out how many cubic yards of soil will be in a truckload, and what that truckload of soil will cost. After that you can figure how far you want to extend the lawn based on what it would cost.

Excellent advice above. i went through a similar issue (20+ trees and 35+ bushes removed on .3 acres). we didnt have to grade the entire area but stil was a pain and still dealing with some issues. i suggest adding a very thin layer of compost on top of the topsoil. the areas we seeded that had the compost seeded much better than topsoil only. and even if you get uneven spots you can add up to an inch a month of soil on top of the grass to even it out later.

I'm not sure what 'good brown soil' is, but topsoil is what you want.

This is a job for a licensed landscaper, not a DIY. They will bring in a tractor with a box blade and as much soil as needed to profile the yard the way you want it. Get several estimates. Assuming they have all the access they need for the tractor, they can do the job in a day...possibly one morning. If they estimate it will take a week, then they are using the wrong equipment. For example using a bobcat or skid steer it would take a week, because just because they have box blade attachments, does not make them capable of doing the job.

It's the geometry of the tractor and box blade on the back that makes it ideal for this type of work. You'll never see a farmer plowing with a bobcat.