Oh come on guys, I did this as an engineering intern when I was 16. It's not like it's rocket science.
Take a set of plans.
Draw a grid on it. Note the width of the grid you drew. Square the length of one side to determine grid area. Kids algebra.
Average the existing elevation at the for corners of a grid box. Write this number down in red pen in the middle of the grid box. More kids algebra.
Average the proposed elevation at the four corners of the same grid box. Write this number down in blue pen. More kids algebra.
Subtract the elevations. This gives you the average elevation difference for the grid box. Multiply by the grid area and you get an approximate volume difference between the existing finished grade and the proposed finished grade, for that box. Make all cuts negative and fills positive (or visa versa, whatever, just be consistent).
Now do it again and again for each grid square and add them up. Denser grids give greater accuracy. Certain grid orientations may provoke more errors, but a tight enough grid always gives you an answer that's close enough to what you need. Determine that through experience.
That's how you do volume takeoff, the old fashioned elbow-grease way. Have fun. That's not all you'll need to know, though. You have to take into account topsoil depth, because you can't backfill with that. Or maybe you can, if you can stockpile it, but only in landscaped areas, and maybe only to certain depths, so plan all that out. You have to take into account shrinkage and swell factors for different soils, which act like multipliers to the calculated volumes, but perhaps only for the cuts and/or fills. You have to take into account depths of rock if you have to rip it or blast it. Fun times. You have to take into account whether certain material is suitable fill, according to your geotech. And then if you're doing a construction cost estimate, you have to take into account how you're doing the cutting and/or filling, what machinery you're using, whether you have any haul-off or import, etc. For all that mess, I'd recommend learning how to use the Means Manual, or better, compiling a detailed list of the quantities of all your previous jobs, and what those cost, and using that as your indexing method to determine future costs. Or even better, hire an engineer.
There are other ways to do this, by the way, such as if you had road sections or other starting info to go off of. And most everybody nowadays uses a TIN in a CAD model. Grids are just the easiest method to explain.
Hydrology, Drainage Analysis, Flood Studies, and Complex Stormwater Litigation for Atlanta and the South East -